You can have three personal loans at once. There is no official limit on the number of personal loans you can have at the same time. It’s common for lenders to only make one or two loans to the same borrower, but there’s no reason you can’t apply for personal loans at more than one lender.
Having said that, it can be difficult to obtain multiple personal loans, as you’ll need to demonstrate your ability to make the payments on all of your personal loans every month, and lenders might be concerned if your existing level of unsecured debt seems excessive.