I think what happens if you have a normally closed to ground switch relying on the pullup resistor it will not work because the LED is pulling the pin down more than the internal 20k pulls it up.
That is probably correct. What is happening is that the LED and series resistor and internal pull-up resistor (if pin is set to input and high is outputted to set pull-up) are acting like a voltage divider. You could measure the voltage at the pin to see what it is, but it's bound to not be a legal logic voltage level. It's pretty problematic to utilize pin 13 as a input to a switch without risking some kind of damage in certain situations.