The account I had linked for Paypal does not have an overdraft yet the bank happily let them take the payment, should they of not just refused it due to insufficient funds at the time?
Most accounts will have the option of an unarranged overdraft available, simply meaning you'll get charged a lot more than you would if it was an arranged overdraft. Depending on the bank this may be a higher interest rate, or a daily rate, just depends on what your accounts T&C's contain.
The only accounts that generally don't have that are young persons accounts.