It's possible/likely that the keyboard (keyboard = the keys and electronics that sense them, not the whole unit) decoder cannot handle more than "n" keys depressed simultaneously, where "n" is what you need to get the polyphony.
Keyboard multiplexing/decoding has always been a limiting factor in low-end synths.
Don't most cheap ones allow you to record in one voice/instrument, then switch to a different voice while playing back the original so that you can "accompany" yourself?
Tim.