I have 3500.00 in unpaid medical bills. These are currently in collections. I recently began making payments of 50.00 per month. I missed last month. Now they have demanded 500.00 per month. They will temporarily allow me to pay 250.00 per month for 3 months. I told them that would be impossible. They said I could continue to send in any amount. My question is, can they sue or garnish my wages if I am paying them what I am able to each month, if they continue to cash my checks?

Nope, if they cash it they are accepting it as payment. I would try to pay a decent amount so they would see that I am trying.

If you are making a good faith attempt to pay what you can, such as $50 per month on a regular basis, it's unlikely that they'd prevail in a lawsuit or obtain any kind of judgement against you. Of course, save all canceled checks.