Yes, FAT formatting would definitely prevent it from seeing it as a viable drive to back up.
Also, be aware that your Time Machine backup drive needs to be at least as big as the two drives combined -- so, if you have a 250GB internal drive and a 500GB external drive and you wish to back up both with Time Machine, your Time Machine drive will need to be at least 750GB in size -- regardless of how much space is actually used on those drives.