The file extension is always the same across all devices - .PDB. The only reason HanDBase for Android would skip a database file with a PDB extension is the following:
1. It's not a HanDBase PDB file. Not likely in this case!
2. It's not a valid HanDBase 3.0 or higher format file. This could be due to a corrupt file or it being an older pre HanDBase 3.0 format. Since the Palm version you had was likely 4 or higher to run on the Pre, then this latter is not likely either.
The next best step is to send one of the databases that does not show on your Android phone to support@ddhsoftware.com. Brian can test to see if it shows on his Android device and we can take it from there..