Wouldn't it be great if this question was simply answered with "Yes!"? Profit-Making Institutions
This brings up an issue that many people know, but may not keep in mind: banks are profit-making institutions. Banks are borrowing money from you and then lending it out to others; they take the loan payment, pay you your interest, and keep the difference. So their incentive is to pay you the least amount they can, and to charge the most they can for loans, which may also be to you. (Find out how to get the bank to pay you for using their services, not the other way around. Read Cut Your Bank Fees.)
Trust Banks to Make Profit
Furthermore, banks were allowed to combine with investment firms and insurance companies in 1999, due to the Financial Modernization Act. So, it's also in a bank's best interest to sell you a variety of products, some of which you may not really need, at the highest price possible. This is true for all profit-making firms, but for some reason many people lose sight of this incentive with banks. Perhaps it stems from the idea that you can trust banks to give you back the money you deposited. But it's clear that you can also trust banks to act like profit-making firms and do what's in their own interest.
So, "Yes!", you can trust your bank, in many different ways.