Occasionally, yes. Usually, no.
Unless you're confident that your debtor's paper bank statement is in his trash with last week's chicken bones and are willing to root around for it, you're not getting that info easily and sometimes, not legally.
While proprietary database information is available for licensed private investigators to locate physical assets like real estate and watercraft, and sometimes employment or businesses, there is no nationwide database of people’s bank account balances.
Because of the Right to Financial Privacy Act, financial institutions will not disclose financial records or account information without: 1) the customer’s consent, 2) a court order, 3) subpoena, 4) search warrant, or 5) other formal demand, with limited exceptions. Furthermore, the Gramm–Leach–Bliley Act (GLBA) passed in 1999, contains several privacy rules restricting financial institutions from releasing customer information.
Investigators advertising they can get bank balances on a consistent basis are probably pretexting for it, which is a legal technique for gathering many types of information but actually prohibited by the GLBA for the purpose of obtaining financial information; or, they have sources in the banking industry willing to violate privacy laws in order to sell this information to them. Consider if you want to risk the possibility of testifying how you and your investigator obtained the information and having it thrown out as evidence, and/or facing other legal implications.
That said, it’s possible to legally locate bank account information as long as it’s already in the public record, like in a divorce filing for example. Or, if the account holder has volunteered the information or been compelled to share it in a debtor examination hearing in court. Meanwhile, the dumpster diving technique mentioned above is still possible, but can be time-consuming and expensive. And more people are opting out of paper bank statements nowadays anyway.
With all this in mind, consider your reasons for wanting to know someone’s bank balance.
If you already have a judgment, you can order the debtor back to court for an examination hearing, where you can ask them about their assets in front of a judge, including their bank account information. You might also consider hiring a private investigator to conduct a bank locate, but keep in mind it will generally not include any balances. However, once you know where your debtor banks, you can then subpoena the bank for records about how much money is in their accounts.
If you’re just trying to find out if it's worth taking legal action, or simply have a personal curiosity, there are other ways to get a general picture of someone’s financial situation.