What are the atmospheric requirements for large dinosaurs?
Oxygen levels have changed greatly throughout Earths's history starting with very little atmospheric oxygen and gradually increasing as photosynthetic activity appeared until all the oxygen sinks such as minerals had been filled, then taking off, reaching as high as 35% of the atmosphere during the carboniferous period 3. The reasons behind this sudden increase in O₂ has to do with the development of bark covered wooden plants, a feature made possible by the molecule lignin, which strengthens the trunks of trees and makes tall plants possible. However, lignin was hard for microbes to digest, and the dead trees built up, never returning their carbon to the atmosphere as CO₂. So the oxygen they had made while alive stayed in the air 4.



The plot shows the upper (red) and lower (green) estimates for O₂ concentration over time. The data for this plot was taken from a review of analysis of sulfur, carbon, iron, and manganese deposits 2. If you're interested in how isotopic analysis can give information about atmospheric oxygen, read that paper, it's detailed.

That spike in the plot is the carboniferous. This period in history is interesting because of the appearance of large invertebrates, such as 3 foot wide dragonflies and 6 foot long centipedes 5. Since these animals move oxygen through a series of tubes called trachea, where air just moves through by diffusion, it was speculated that the higher concentration of oxygen allowed this growth. Recent research indicates that insects can force air through their trachea, and that competition between predators and prey could have explained most of the size 6.

So how does this affect dinosaurs? The carboniferous period was about 350-300 million years ago, which corresponds with the spike on the graph. Dinosaurs appeared in the Triassic, which was about 250-200 million years ago, but oxygen levels were at 16% of the atmosphere at the time 7 ( current O₂ levels are about 20% ) If we move into the Jurassic, about 200-150 million years ago, oxygen levels rise to about 26% 8. In the Cretaceous, about 150-65 million years ago, O₂ was at about 30% 9. The O₂ percentages are taken from the infoboxes on the right side of the wikipedia articles, which in turn take the number from a plot very similar to the plot I posted here, despite being in German 10.

I compiled lists of heaviest and lightest dinosaurs by going through the wikipedia article on dinosaur size 11 and going to the article on each dinosaur and checking if it was Triassic, Jurassic, or Cretaceous.
If we look at the heaviest dinosaurs, most of them appeared during the Cretaceous:

Heaviest Dinosaurs          
Theropods                           Metric Tons     Period
Spinosaurus aegyptiacus             7-20.9          Cretaceous
Carcharodontosaurus saharicus       6.1-15.1        Cretaceous
Giganotosaurus carolinii            6.1-13.8        Cretaceous
Tyrannosaurus rex                   6-9.5           Cretaceous
Oxalaia quilombensis                5-7             Cretaceous
Acrocanthosaurus atokensis          5.6-6.2         Cretaceous
Tyrannotitan chubutensis            4.9-5.6         Cretaceous
Suchomimus tenerensis               2.7-5.2         Cretaceous
Therizinosaurus cheloniformis       5               Cretaceous
Tarbosaurus bataar                  4-5             Cretaceous

Sauropods                           Metric Tons     Period
Amphicoelias fragillimus            122.4           Jurassic
Argentinosaurus huinculensis        60-90           Cretaceous
Antarctosaurus giganteus            69-80           Cretaceous
Mamenchisaurus sinocanadorum        75              Jurassic
Sauroposeidon proteles              40-60           Cretaceous
Paralititan stromeri                59              Cretaceous
Brachiosaurus altithorax            28.7-56.3       Jurassic
Puertasaurus reuili                 50              Cretaceous
Ruyangosaurus giganteus             50              Cretaceous
Alamosaurus sanjuanensis            50-60           Cretaceous
Futalognkosaurus dukei              38.1-50+        Cretaceous
Turiasaurus riodevensis             40-50.9         Jurassic/Cretaceous
Camarasaurus supremus               47              Jurassic
Diplodocus hallorum                 30-42.5         Jurassic
Supersaurus vivianae                32-40.2         Jurassic
Elaltitan lilloi                    42.9            Cretaceous
Tehuelchesaurus benitezii           41.3            Jurassic
Apatosaurus louisae                 18-41.3         Jurassic

Ornithopods                         Metric Tons     Period
Shantungosaurus giganteus           9.9-22.5        Cretaceous
Iguanodon seeleyi                   15              Cretaceous
Saurolophus angustirostris          6.6-9           Cretaceous
Iguanodon bernissartensis           8.3-8.6         Cretaceous
Edmontosaurus annectens             3.2-7.6         Cretaceous
Brachylophosaurus canadensis        4.5-7           Cretaceous
Saurolophus osborni                 6.6             Cretaceous
Lanzhousaurus magnidens             6               Cretaceous
Parasaurolophus walkeri             3-5.1           Cretaceous
Charonosaurus  jiayinensis          5               Cretaceous
Barsboldia sicinskii                5               Cretaceous

Ceratopsians                        Metric Tons     Period
Triceratops horridus                5-14            Cretaceous
Triceratops prorsus                 6.5-11          Cretaceous
Titanoceratops ouranos              6.5-11          Cretaceous
Eotriceratops xerinsularis          10              Cretaceous
Pentaceratops sternbergii           3-4.8           Cretaceous
Pachyrhinosaurus canadensis         3-4.4           Cretaceous
Styracosaurus albertensis           4.2             Cretaceous
Agujaceratops mariscalensis         2.6             Cretaceous
Centrosaurus apertus                1.1-2.5         Cretaceous
Coronosaurus brinkmani              2               Cretaceous
Rubeosaurus ovatus                  2               Cretaceous
Achelousaurus horneri               2               Cretaceous
Pachyrhinosaurus lakustai           2               Cretaceous
Chasmosaurus belli                  2               Cretaceous

Thyreophorans                       Metric Tons     Period
Dacentrurus armatus                 5-7.4           Jurassic
Ankylosaurus magniventris           1.7-6           Cretaceous
Stegosaurus stenops                 2.6-5.3         Jurassic
Cedarpelta bilbeyhallorum           5               Cretaceous
Hesperosaurus mjosi                 3.5-5           Jurassic
Tuojiangosaurus multispinus         4.8             Jurassic
Wuerhosaurus homheni                4               Cretaceous
Niobrarasaurus coleii               4               Cretaceous
Stegosaurus ungulatus               3.5             Jurassic
Gobisaurus domoculus                3.5             Cretaceous
Nodosaurus textilis                 3.5             Cretaceous
Palaeoscincus costatus              3.5             Cretaceous
Sauropelta edwardsi                 3               Cretaceous
Edmontonia rugosidens               3               Cretaceous
Edmontonia schlessmani              3               Cretaceous
Edmontonia longiceps                2.3-3           Cretaceous
Tuojiangosaurus multispinus         2.8             Jurassic
Euoplocephalus tutus                2-2.7           Cretaceous
Jiangjunosaurus junggarensis        2.5             Jurassic


This isn't exactly a fair way to do it, but I didn't want to calculate average dinosaur mass over time. I know that some groups, such as the ceratopsians didn't appear until the late Jurassic, skewing the results, and I didn't make any attempt to differentiate between late or early periods. However I also noticed that many of the smallest dinosaurs appeared in the Cretaceous as well. Pay attention to the units, even the smallest sauropods are measured in metric tons.

Lightest Dinosaurs
Theropods                           kg              Period
Parvicursor remotus                 0.137           Cretaceous
Epidexipteryx hui                   .164-.391       Jurassic
Compsognathus longipes              0.26            Jurassic
Ceratonykus oculatus                0.3             Cretaceous
Juravenator starki                  0.34-0.41       Jurassic
Ligabueino andesi                   0.35            Cretaceous
Microraptor zhaoianus               0.4             Cretaceous
Sinosauropteryx prima               0.55-0.99       Cretaceous
Rahonavis ostromi                   0.58            Cretaceous
Mahakala omnogovae                  0.76-0.79       Cretaceous
Xiaotingia zhengi                   0.79            Jurassic
Mei long                            0.85            Cretaceous
Microraptor gui                     0.95-1.50       Cretaceous

Sauropods                           Metric Tons     Period
Pleurocoelus nanus                  0.5             Cretaceous
Magyarosaurus dacus                 0.75            Cretaceous
Europasaurus holgeri                0.8-1           Jurassic
Bonatitan reigi                     1               Cretaceous
Lapparentosaurus madagascariensis   1.4             Jurassic
Lessemsaurus sauropoides            1.8             Triassic
Lirainosaurus astibiae              1.8-4           Cretaceous
Shunosaurus lii                     2.2-6.7         Jurassic
Ampelosaurus atacis                 2.5             Cretaceous
Amargasaurus cazaui                 2.6-3.8         Cretaceous
Hypselosaurus priscus               2.7-8           Cretaceous
Euhelopus zdanskyi                  3.4             Cretaceous
Neuquensaurus australis             3.5-6.1         Cretaceous
Rinconsaurus caudamirus             4.1             Cretaceous
Atacamatitan chilensis              4.3             Cretaceous
Dicraeosaurus hansemanni            4.4-5           Jurassic

Ornithopods                         kg              Period
Gasparinisaura cincosaltensis       1-13            Cretaceous
Yueosaurus tiantaiensis             3.9             Cretaceous
Fulgurotherium australe             6               Cretaceous
Notohypsilophodon comodorensis      6               Jurassic
Yandusaurus hongheensis             6.6-7.5         Jurassic
Hypsilophodon foxii                 7-21            Cretaceous
Thescelosaurus sp.                  7.9-86          Cretaceous
Valdosaurus canaliculatus           10              Cretaceous
Haya griva                          11              Cretaceous
Agilisaurus louderbacki             12              Jurassic
Drinker nisti                       20              Jurassic
Changchunsaurus parvus              20              Cretaceous
Qantassaurus intrepidus             20              Cretaceous
Zephyrosaurus schaffi               20              Cretaceous
Oryctodromeus cubicularis           20              Cretaceous
Orodromeus makelai                  20              Cretaceous

Ceratopsians                              kg        Period
Liaoceratops yanzigouensis                2         Cretaceous
Yamaceratops dorngobiensis                2         Cretaceous
Psittacosaurus sinensis                   4.1       Cretaceous
Psittacosaurus lujiatunensis              5         Jurassic
Yinlong downsi                            5.5       Cretaceous
Micropachycephalosaurus hongtuyanensis    5.9       Jurassic
Chaoyangsaurus youngi                     6         Jurassic
Xuanhuaceratops niei                      6         Jurassic
Bagaceratops rozhdestvenskyi              7         Cretaceous
Psittacosaurus meileyingensis             8         Cretaceous
Psittacosaurus neimongoliensis            8-8.4     Cretaceous
Archaeoceratops oshimai                   10        Cretaceous
Psittacosaurus mongoliensis               12.1-20   Cretaceous
Psittacosaurus sibiricus                  15        Cretaceous

Thyreophorans                       kg              Period
Scutellosaurus lawleri              3               Jurassic
Emausaurus ernsti                   50              Jurassic
Scelidosaurus harrisonii            64.5-270        Jurassic
Animantarx ramaljonesi              300             Cretaceous
Struthiosaurus transylvanicus       300             Cretaceous
Struthiosaurus austriacus           300             Cretaceous
Gargoyleosaurus parkpinorum         300             Jurassic
Mymoorapelta maysi                  300             Jurassic
Minmi paravertebra                  300             Cretaceous


So what does this mean for the relationship between oxygen levels and dinosaur size? Higher oxygen levels in the Jurassic and Cretaceous probably allowed for larger animals to exist, however as time went on Dinosaurs became more diverse in general, including size. Many of them evolved to become smaller, despite being allowed to become larger by higher levels of oxygen. We should also note that even though oxygen levels were still lower than they were during the carboniferous, the animals were much larger, most likely due to their more efficient oxygen transport systems (lungs, 3(4?) chambered hearts, sealed circulatory systems, hemoglobin).

If we think about what could drive species to become larger, competition for food and protection from predators comes to mind. Sauropods that could reach the leaves in taller trees had distinct advantages over those who could not. Prey animals stood a better chance at fighting off predators if they were bigger. Likewise, predators could kill larger prey if they became larger too. Oxygen didn't drive them to become larger, it simply allowed them to.

To actually answer the question, could the modern atmosphere support animals as large as dinosaurs? It probably could. The heaviest living animal today is the blue whale, at an average mass of 110 metric tons, is heavier than all but 1 of the sauropods on the list I compiled. The heaviest land animal living today is the African Elephant, with average mass of 4.9 metric tons and as high as 10 metric tons, would be a respectable competitor with many of the dinosaur groups I listed 12.

Another interesting question is not how oxygen affected the size of organisms, but how it affects it their metabolism. Cold blooded animals don't need as much oxygen because they don't spend as much energy. Warm blooded animals must constantly expend energy to maintain body heat, and this requires greater consumption of oxygen. Obviously being large and warm blooded would force you to need even greater levels of oxygen than 1 or the other. It may be that the advantages of being warm blooded outweighed the advantages of large size, but I don't want to dig into that question right now.