If you get into a hammock, you can fall out of it.

The only way I can picture someone falling a hammock is if the hammock is on the ground and the idiot is standing on it.

You can lie in a bed and fall out of it.

You can lie on a bed and off it.

If you tell me you are lying in bed, I imagine you with blankets, sheets, etc. 

If you tell me you are lying on a bed, I imagine you on top of the bed with or without a blanket.  However, you are certainly not between the sheets.  Perhaps you are just resting.