Is there a limited amount of money that a savings account should have

In short, there is no limit on the amount of money that you can put in a savings account. No law limits how much you can save and there’s no rule stating that a bank cannot take a deposit if you have a certain amount in your account already.

The only limits imposed on the balance of your savings account are the limits imposed by the bank itself. Very few banks impose a limit on your savings account’s balance. Banks make money when you deposit your money into an account, so you’re unlikely to ever have a bank turn your deposit away.

If you do run into a situation where a bank will not accept further deposits, there’s nothing stopping you from opening an account at another bank.

In fact, you could open savings accounts at multiple banks if you feel like it. This could be an important thing to do for the security of your savings depending on how much you have.