Is it possible that atoms with 120 protons are possible, but that atoms with 119 protons aren't possible?
According to the Wikipedia page element 119 synthesis has been attempted indicating physicists think it is possible.  There is some speculation as to whether our current technology could even detect such short lived elements such as elements 119 or 120.  There is also speculation that the technology today may not be able to produce element 120.  It may be possible for such gaps (who knows maybe element 562 is stable and element 561 cannot form?) but at the current time no gaps are known with certainty.

One of the bigger problems for higher atomic number atoms is that as the nucleus becomes more charged the electrons orbiting the nucleus must go faster and faster to stay in orbit.  However nothing can exceed the speed of light and thus if an atom were to have electrons near the speed of light (say 99.99999% of c) at rest. The mere revolving of the earth around the sun would require the electron to move faster than light which presents and interesting case/thought experiment. Because of this it is reasonable to deduce that there is a penultimate element atomic number where anything larger would required electrons to exceed the speed of light though I cannot say what the magic number would be.

I will say though your question has a quite lot of merit as the Oddo-Harkins rule holds that the abundance of elements with odd atomic numbers in the universe is far less than that of elements with even atomic numbers except for hydrogen of course.  Further radioactive nuclei with even atomic numbers tend to be more stable and have a longer half life than those with odd atomic numbers.