Inbred mice has no severe phenotype outcome?
Some information on inbred strains of laboratory mice:

https://en.wikipedia.org/wiki/Inbred_strain

http://what-when-how.com/molecular-biology/pure-line-molecular-biology/

A relevant quote on the consequences of inbreeding:


  Inbreeding in allogamic organisms bring the deleterious recessive alleles to homozygosity; the immediate consequence is an increase in the frequency of defective offspring, or, in another words, an increase in the genetic load of the population. This phenomenon is called inbreeding depression or inbreeding degeneration. As inbreeding continues, the deleterious alleles are selected out and eventually disappear. The original heterozygous populations are often more fit than the resulting pure lines because they profit from heterosis and balanced polymorphisms; the main advantage of pure lines is the quick production of many individuals with the same well-adapted genotype, while the allogamy continuously generates new genotypes.


In other words, inbreeding is harmful because it makes it more likely that offspring will have two copies of bad recessive alleles, meaning those alleles get expressed, meaning the organism gets the bad consequences that wouldn't show up if they had only one copy. This is what happens over a single generation however; over many generations the bad alleles are selected against, precisely because they are harmful to the organism so it reproduces less, and after enough generations of inbreeding you hit a point where all individuals are genetically identical and have all the "good" alleles (if they were lucky; otherwise they die out or stay stuck with some bad-but-not-fatal alleles). They still may be worse off than their more diverse ancestors, but they're not completely messed up like their unfortunate great-aunts and uncles who didn't make it either.

The big difference between inbred mice and dogs or tigers is the "for specific phenotype" aspect. Laboratory organisms are inbred so that you get a large pool of genetically-identical individuals, meaning they're much easier to experiment on. The aim is the inbreeding itself, not any particular phenotype. For example if you look at the page for the most popular strain of laboratory mice, C57BL/6, you can see it has many different properties and is used for many different things.

On the other hand, dogs aren't inbred for the purpose of inbreeding or of being genetically identical; the aim is to get desirable phenotypes, and inbreeding is just an efficient way of achieving that aim. It also isn't obvious that many of the problems purebred dogs have is inbreeding (i.e. lack of genetic diversity, high levels of homozygosy) per se, but the fact that the traits being bred for are just plain unhealthy for the dog, or part of a bell curve that include bad outcomes at the edges. For example, Syringomyelia in the Cavalier King Charles Spaniel:


  Some researchers estimate that as many as 95% of CKCSs may have Chiari-like malformation (CM or CLM), the skull bone malformation believed to be a part of the cause of syringomyelia, and that more than 50% of cavaliers may have SM.* It is worldwide in scope and not limited to any country, breeding line, or kennel, and experts report that it is believed to be inherited in the cavalier King Charles spaniel. CM is so widespread in the cavalier that it may be an inherent part of the CKCS's breed standard.


(emphasis mine)

Same thing for that spine malformation that's related to selecting for corkscrew tails. The genes that make the tail corkscrew also mess with the spine.

In other words, the issue isn't inbreeding or not but whether the genes themselves are harmful. When organisms are selected for traits that are directly harmful in their extreme, or are associated with harmful genes that just happen to be next to those that are selected for in the chromosome, then the harmful consequences will spread through the population. Inbreeding is only a problem insofar as it allows the process go faster (more offspring per generation have the desired trait). On the other hand when you're just inbreeding with no specific focus on phenotype, or not phenotypes that have obvious harm associated (i.e. no lab would select for a frivolous trait that also causes harm. They're either selecting for the harmful trait on purpose, or they're selecting against it, because they'll want animals that are as healthy as possible except for the one variable they're interested in), then you'll end up with populations that are fairly normal except for some of the direct consequences of genetic uniformity.

It should be noted that most purebred dogs probably aren't inbred strains the same way many laboratory animals are; those are genetically identical, so the whole point is that their offspring will be like they are. So while they may be less fit than a non-inbred version of them might be, their offspring won't be any less fit than they are. And this is not what's observed with purebred animals like dogs and horses; individuals aren't identical, and looking at the page on Syringomyelia it seems the problems are getting worse.