That is certainly a possibility. But the situation is much more reasonable if you imagine that the doctors ran different tests.
For example, suppose we have a prior probability of: Tumor - .49, Concussion - .49, meningitis - .02.
The first doctor performs an MRI that conclusively proves that there is no Tumor. He now conludes that there is a 96% chance of concussion, and 4% chance of meningitis.
The second doctor performs a cognitive test that conclusively proves there is no concussion, but a 96% chance of tumor, and 4% chance of menigitis.
If we combine these two tests, we have disproven the two post likely causes, so the less likely cause becomes reasonable.
This doesn't work out quite so nicely in reality because of false negatives and (as you point out) the non 0% chance of an unspecified disease. Also, ideally you would tell the second doctor about the first test.
