Yes, you can have more than one card from a single issuer. However, the extra account carries both pros and cons.
A few situations could warrant opening a second (or third) credit card with the same bank. For instance, you may have started a relationship with your bank at a time when you only qualified for a high APR, no-frills card. Years of on-time payments can earn you an invitation for a rewards credit card or for an account with a higher credit limit.
Likewise, your current bank may now offer an account with features and benefits that didn't exist when you signed up for your current card. Under federal rules, your bank can't just change the terms of your existing account -- you'll need to get approved for an entirely new account.
Opening a second account with the same bank, especially if your issuer offers you a no annual fee card, can also help improve your credit score. Banks play close attention to your credit utilization -- the percentage of your available credit that you actually use. If your bank offered you the account without requiring a "hard pull" of your credit report, you'll likely maintain your credit score instead of sustaining a short-term hit tied to a new credit application.
On the other hand, that additional account includes a few drawbacks. First, you generally won't be able to transfer a balance between credit cards from the same issuer. That could get frustrating if you're stuck with a balance on your original account that you can't surf to your lower APR credit card.
Also, a problem with one of your accounts could impact all of the accounts at the same institution. For instance, missing a payment with one account could trigger a penalty APR or a reduced credit line on the account in good standing. Likewise, cases of identity theft or card skimming could lock you out of all of your accounts while your bank investigates the issue.
Finally, the same credit limit increase that helps your overall credit score could scare off lenders concerned about your available unsecured credit. While you still may earn consideration for a mortgage or a home loan, you might want to wait a few months before responding to an instant approval credit card offer from a different bank.
Overall, picking up a second account from the same bank can be useful if you're getting more value than the cost of maintaining the account. Just keep a secondary credit card from a competing lender as backup in case of emergency.