As mentioned, there are a lot of different forms of CMT, but that is a general term you can use for hereditary neuropathies. Another term, less used I think, is HMSN, "hereditary motor and sensory neuropathy". As also mentioned, there's all sorts of different sub-types, and there is some overlap between the two different systems. (Kind of like the Dewey Decimal system and the Library of Congress system for organizing books.) Here's a brief article,
emedicine.medscape.com/article/1173104-overview

In my case, my doctor put in a pre-authorization for a genetic test and it was rejected by my health insurance. But the test is from Athena Diagnostics, who has a program by which they guarantee that the patient's out of pocket expense is limited to 20% of the cost (if you agree to the program and send your payment within 30 days of the test). So, the next time I saw him, he just sent in for the test. I had my blood drawn, received paperwork from Athena about the program, and paid them my 20%. I've received the results (MFN2: negative). I have no idea yet what my insurance company is going to say, but it's not my problem -- a piece of the paperwork allows Athena to argue with the insurance company on my behalf, to convince them to pay.

My insurance company rejected it because the result wouldn't make a difference in my care or in the outcome of the disease -- and I agree that it wouldn't, but it would give me some peace of mind and some understanding of what's causing my symptoms. It's still expensive, tho -- the test cost was $940 and so my 20% was $188.

Athena is the leading provider of genetic tests for CMT and hereditary neuropathies. They have vast panels that will test for everything they can, but my physician has been pretty focused on specific tests relate to my symptoms. You might ask your doc if there are some specific tests from Athena that would be appropriate (so that your 20% doesn't go through the roof).

Here's the scoop on the "Access Athena" program,
athenadiagnostics.com/content/ordering/access-athena

And their main web site is here (you can browse the tests they've got),
athenadiagnostics.com/

(You'll have to copy and paste the links since neurotalk doesn't trust me to post links yet.)