If you have opened two cash ISAs in the same tax year with different banks you will be asked to close the second one. The Inland Revenue will eventually realise what has happened because they can check every account as all the banks etc have to report all transactions each year. It is unlikely that you will get a fine especially if you go to the second bank and say that you made a mistake. If they are equity ISAs the same applies except that you are normally allowed to keep the second one invested but without the tax benefit.