Indeed it is, unless of course you've been a less than stellar client, meaning overdrawn, late payments, below balance minimums etc. If you've run into no problems, have been their for a bit, your bank would without a doubt be the easiest, & fastest (meaning less paperwork hassle, background etc.)
It is easier to get approved for a credit card from a bank if you have a bank account in that bank. This is because the bank knows your financial standpoint as well as your financial history. It is also possible to get from other banks although the process may take longer.