I need a bit more information and pictures are always a plus. We have vine borers that can get into the base of the plant as it enters the ground. Not sure if this is the type of damage you're talking about.
Please give me more information.
Now I'm wondering if it is an insect problem. However, whole sections of the vine are missing right up to the cantaloupe but the leaves are left.
It looks like they're just being snipped off and not eaten. This is typical of rabbits. As you may know, their teeth grow throughout their lives and they are constantly trying to find something to snip. Although anything is possible. I don't think it's an insect problem however.
Sprinkle some bloodmeal around the perimeter of your melons. Rabbits don't like to cross over that. It has to be reapplied after a rain.