How do I measure the thickness of something with a lip or frame?
There are special forms of Vernier caliper such as these known as neck calipers.





  These neck calipers enable measurements in hard to reach places.
  They offer you the following benefits:
  
  
  You can measure wall thickness inside bores and recesses.
  




Obviously, the machinists calipers in A.I.Breveleri's answer will be cheaper and will probably work for a much wider range of situations.



Another type is the Curre-jaw caliper - again with a vernier scale - though both these jaw types are available on calipers with electronic digital readout





There are also a very wide range of external dial calipers, as kris and pcdev identified.