Can the bank claim money from a personal account to cure an overdraft?
The answer will be governed by contract law principles. I suggest your read the terms carefully. It is not uncommon for such agreements to contain a security clause, allowing the bank to attach other deposits.
The following is an example clause:
If you ever owe us money as a borrower, guarantor or otherwise, and it becomes due, we have the right under law (called “setoff”) and under this Agreement, you agree to grant us a security interest in your deposit account and authorize us to use the money from your account to pay the debt. We may use the money to pay the debt even if such a withdrawal results in an interest penalty or dishonor of checks or any other obligation that you have incurred. In the case of a partnership or a joint account, each partner or joint account holder agrees that we may use the money in their individual accounts to satisfy obligations of the joint account or partnership account. You agree that the security interest granted by this Agreement is consensual and is in addition to our right of setoff. However, the right of setoff and security interest may not apply to your account if (i) it is an IRA or a tax-deferred Keogh Retirement Account; (ii) the debt is created by a consumer credit transaction under a credit card plan; or (iii) the debtor’s right of withdrawal arises only in a representative capacity.