You can go to bank x with proposal from bank y, if bank Y can match the offer, you can return to bank X, you can visit as many banks as you want and you can tell them what other banks offered you.
Currently, where I live, if you want to grab a loan to purchase a house, you need to have 20% of the amount you want to lend, before most banks will even consider handing out a loan.
This has only become the standard the last years, hasn't always been that way.