Yes, you can develop neuropathy without being a diabetic. Although diabetes is the number one cause of peripheral neuropathy today, there are also many other causes as well.
Another common cause is heavy metal poisoning, oftentimes found in people whose diet consists of a lot of deep water fish such as bluefin tuna, because of the high mercury content in these fish. People who eat sushi on a daily basis sometimes develop symptoms of neuropathy as well.
Peripheral neuropathy can also be caused by a vitamin deficiency, especially a lack of B complex vitamins in the blood.
Many alcoholics and heavy drinkers develop a deficiency in the water-soluble vitamins such as vitamin B, because they are flushed out of the body as it attempts to rid itself from the tocsin of alcohol.
So if you develop symptoms of neuropathy, the first thing that your doctor will probably do is test your blood sugar levels see if they are the diabetic range.
If he can eliminate diabetes from being the cause, he will run further tests to check for heavy metals, vitamin levels, etc. in your blood which may also be causing the symptoms.