What kind of external drives? Firewire? eSata? Raided? Other? And, how old are the drives in question?
Meantime, could be a number of reasons. A few of the more common:
1. Bad power brick. Swap it out for one that you know works and make sure that it's one that the manufacturer supplied. You usually lose warranty but not using the power brick that came with the drive.
2. Bad connections. Make sure that everything it seated properly.
3. Bad cables. Swap out the firewire or whatever you're using for ones that work.
4. Bad port on the drive. Nothing that you can do about that. It might be able to get fixed at a repair facility.
5. Corrupted file structure. First go to disk utility and run a first aid check. Sometimes a permissions repair will do the trick.
6. A drive about to head south on you. Oftentimes drives don't just cease to function, but give you some advance warning by acting buggy as you describe. Nothing you can do about that either other than getting the drives over to a repair shop asap and seeing if they can copy the data over to a new drive.