LED lighting question
It sounds like the line voltage is low.  I have had this issue before with some cheaper and counterfeit bulbs from China.  Check your line voltage at the socket (the thing the bulb screws into) and see how much of a drop it has from, say, a nearby outlet.  You can then get a general idea of how much voltage drop the 4 LED bulbs are causing in total.  What happens is that the little inverter type power supply board in the LED is really fussy in the cheapies, and if the voltage at the junction box is already low (old house, end of a trunk , other loads in line, etc.), the voltage is too low to run five, but just enough to run 4.  you can sometimes see this if you take a high frame rate HD video (70fps and higher) of 1 led bulb installed, and then with 1 less than the number that is stops working at (5 in your case, so 4 for you).  Then playback the videos on a player at slow speed.  You will see a flicker with 4, but no flicker with just 1.  T322his is due to the driver voltage being just sufficient with 4 lights, but too low with 5.

If you have a dimmer, try all 5 without the dimmer.  If that works, you either need a magnetic driver type dimmer or you have non dimmable bulbs.  If it doesn't work, your line voltage is too low and it simply won't work.  And remember, this is all assuming you have checked for shorts, bridging, etc in the fixture, wiring and sockets.