Can the FBI recover deleted files?

A normal delete does not actually delete a file, it just marks the hard drive space as available for later use. All the data are still there, and there are software programs that can bring it back more or less intact.

Even after a real deletion, or after the disk is reformatted, traces of magnetism remain on the hard drive, and with special equipment it is possible to read those traces and reconstruct what was there before deletion. This is true even after the deleted content has been overwritten once or twice.

It is not only FBI that can read deleted files, there are commercial companies that offer similar services, for example to businesses that accidentally reformatted the hard drive containing all their customer data.

Some programs offer a "safe delete" option which overwrites deleted files 20-30 times with random noise, to make sure that the data are really gone. This is now the standard option for emptying the waste basket in Mac OS X, for instance.

The drawback of "safe delete" is that it is much slower, especially if you delete a lot of data. And of course it means that even the companies that specialize in retrieving deleted data cannot help you if you accidentally delete something you shouldn't.