It seems like the more accurate, but archaic translation would be "I cannot obtain sleep in this bed" or more commonly "I cannot GET to sleep in this bed". At the very least thinking of it in that way will help me to remember to use 'conseguir' instead of 'poder' in this scenario.
Eu não posso dormir nessa cama = im not allowed to... / eu não consigo dormir nessa cama = im not able to.... you dont have the ability to do that. You dont fit the bed, maybe it is not comfortable... whatever, something that makes you avoid sleeping in that bed.
I had no idea of the differences - I assumed posso was like the Spanish puedo, and we would say "No puedo dormir" to say that we aren't able to sleep. Thanks for highlighting this.