I have an H2 and had never tried transferring files to it, so........
First attempt was a wav file at 44.1kHz/16bit and it didn't play ('no file' message). Found out it was a mono (not stereo) file, but not sure if being mono was the reason it didn't play.
Second attempt was a stereo wav file at 44.1kHz/16bit and it played. I had renamed it to MONO-002.wav, but I tried a shortened abbreviation of the original file name 'Space Weasel.wav' to 'Sp-We.wav' and it loaded and played OK. Probably think there may be an eight character limit as Rusty pointed out. Not sure about spaces, so maybe use a dash or underscore rather than a space if renaming.
The H2 manual states it will handle a few formats, so make sure what you import matches......
Quantization
16/24bit
Sampling frequency
44.1kHz, 48kHz, 96kHz (Stereo mode)
44.1kHz, 48kHz (4CH mode)
The tiny screen and menu on the H2 and not great eyesight was a reminder of why I bought an H4n