"The dog cannot get up onto the bed."
Translation:El perro no puede subir a la cama.
No. Because you're talking specifically about the dog being able or not being able to do something.
Now if you're talking about something not being possible in general, then you could use se puede.
For example; "no se puede entrar aquí" would translate to something like "entrance here is not possible."
Similarly you may see a sign like "no se permite fumar" meaning "smoking is not allowed" as opposed to someone commenting about a person by saying "no puede fumar allí" meaning "he/she can't smoke there."