If you have medical insurance at work that pays 80 percent of medical bills can you get additional coverage for the other 20 percent?

Yes and no.

There are supplemental insurance programs available - do an internet search for "supplemental insurance."

Programs such as these will help pay that remaining percentage - in most cases, a large portion of it at least.

Adding supplemental coverage can be a good option. I have an accident supplemental that does exactly what you are asking. It will pay up to $5000 per person per year for any expenses related to an accident only that your primary carrier does not pay. This means it will pay for deductibles, co-pays (like your 20%), extra services (your plan only pays for 6 physical therapy visits and you need 10) etc. Because it only pays what your primary carrier does not it is very inexpensive ($30/month for the whole family).

If you also want coverage in the event of illness you can also by supplemental products for that as well.