The other answers miss out on a few details that can still cause issues.
The resolution to most of the save game bugs in X-Com: Enemy Unknown lie with the file profile.bin, and what the game tries to do when it thinks there is something wrong. Some of the self-repair features can kick in and recreate the problem if you do not follow all of the following steps.
With the game turned off, open the save game location (C:\Users\%username%\Documents\my games\XCOM - Enemy Unknown\XComGame\SaveData by default)
Copy any files you wish to keep to a different folder.
Delete the entire folder.
Once the game reaches the main menu, it will briefly say "Loading Profile".
At this point, you can copy the files you moved earlier back into the save game folder.
Bypassing any of these steps can cause issues. If the save game folder is deleted, though, the game starts from scratch, creating a brand new profile.bin and SaveData folder instead of trying to recreate it with data from Steam and other hidden files. Any old saves dropped into the folder will be read and added to the new profile, and any new saves will work as normal--until you reach 90+ saves again, that is. It is a particularly frustrating bug because it is easy to seemingly fix the game and have it come back a launch or two later.