Is it the insects eating the roots or is it their grubs? A common cause of root destruction is vine weevils. which look like little black beetles. Their curved, whiteish grubs do the damage but you may see the adults, which only eat the uppper part of the plant. You can buy stuff like Provado to kill the grubs and you have to pick off the adults by hand, preferably at night. Another common cause of root destruction can be ants, as sotongeoff has pointed out, but I'm sure you'd haved recognised them.
Vine weevils are a horrible pest and you sometimes have to be quite persistent to get rid of them. They particularly like the soil in pots, because it is easy for them to dig down in it to lay their eggs. Hope this helps.