Negotiating a rental fee is really dependent upon the owner of the property that's being leased.

If you are renting someone's personal income property (two family, second home, etc.), there can be some room for negotiation if the landlord thinks you'll take excellent care of his investment, have good references, and good credit. Generally, they can be very helpful if they see the tenant as a long term asset. If they really like a potential tenant, the LL may even be willing to let the tenant pay the security deposit off on time. A good tenant, one who pays their rent on time and treats the apartment as their own home, is a gem to a LL.

When the property being leased is part of an apartment building or complex owned by an organization, it's a different situation. Then it's just business - the rent is the rent, the security is the security. If you can't afford the two bedroom, you had better ask to see the one bedroom.