Well, you don't give a lot of information but I will try to answer your question. It depends on who the bank is that is taking the money out of your bank account. If it is the bank where you have your account and it is taking money out of your account which it claims you owe it, then the answer is, yes. It can. Why? Because when you openned your account with the bank you signed a contract stating that they can take whatever they claim you owe them from whatever funds you have deposited with them. This is how they can take penalties and fees out of your account, whether you agree or not, because you gave them permission to when you openned the account.
Now if the bank which is taking the money out of your bank account is NOT the bank which holds your account, then the only way they can do that is with a writ of garnishment from a court. You would know if you have had a writ of garnishment issued against your account because they would have had to serve you with the papers and documents. I assume this is not the case. If the bank you allegedly owe the money to is the bank that has your account, then the remedy is simple. Move your account to another bank that you do not owe money to. But make sure you do not inform the first bank which bank you are moving your account to.
Regarding the attorney's trust account. It is not used by "people with a lot of money to protect their money". The lawyer's trust account is just as susceptible to a writ of garnishment as any bank account, if the creditor knows the money is there and belongs to you, then the creditor can garnish the trust account to the extent of the money it holds belonging to you. Of course, they cannot find out that your money is in the trust account unless you tell them, because the lawyer cannot. It is privileged information.