Try the following and see if it will help. NOTE: Be really careful. You do not want to damage the phone. Also, if the phone is under any kind of warranty, then I would advise against opening it up as it would void the warranty.
Sometimes the pins in the phone (the little gold prongs inside the ports) can get dirt, dust, food, etc., in them and need to be cleaned out, or they get bent, flattened and/or broken.
Try opening your phone to get a good look at the SIMs contacts and check for dirt/food//damage, etc. If they are too flat you can try lifting them up (extremely gently, as they are fragile) with a small toothpick/tool. You can also try cleaning them with an electronics cleaner (something high in alcohol content so won't leave...