I have just changed my switches and plugs and have found that one plug on an outlet is showing open neutral. The problem plug is the one that is controlled by a switch. When the switch is off my tester shows "open neutral" but when the switch is on it show "correct". The other "hot" plug on the outlet shows "correct" always. I have recheck all of the neutrals on the circuit and they all are tight and connected. What else can I check and how dangerous is this. Thank you.

Sounds like the neutral is switched.. Usually the white wire is neutral and black is power.. if the switch is suppose to turn the outlet on and off and the tester is a plugin type (granted there are many types of testers) when the switch is off the outlet should show nothing when tested.. recheck the switch and see if the white wire or black is being switched.. It should be the black or power..

Also check white wire is on white screw, and hot on dark screw.