This comment confused me on first reading. In Australia, it's very much common that you bring your own bed (and other furniture) with you from place to place. (You can get furnished apartments, but they're much less common, at least in Sydney.)
Not the OP, but in the UK(well, London at least) it's very common to have furnished apartments and not to lug your furniture every time you move.
Access to many apartments and houses in London is very restricted. You'd have to take your furniture apart to get it through the door and up the staircases.
That said, a mattress will fold and bend round corners easily, so I don't think this is a big issue if you're renting. (Getting rid of the landlord-provided mattress may be a bigger problem in my experience)
Apartments in the US usually come with heavy appliances such as refrigerators and ovens but not furniture. Furnished apartments are also much less common here.
Are you in the US, or somewhere else?