Thousands of people are cramming in tests, elective procedures and specialist visits before year’s end, seeking out top research hospitals and physician groups that will be left out of some 2014 insurance plans under the new health law, health-care providers say.

Many insurers offering plans under the law are slimming down their networks of doctors and hospitals in a bid to lower the cost of policies, which begin coverage Wednesday.

Health insurers are especially focused on paring academic teaching and research hospitals from their networks because they generally charge more than community hospitals for similar services.

Ben Rosenthal, a 57-year-old freelance market researcher in Los Angeles, rushed to squeeze in a lung exam the day after Christmas. His new insurance plan under the health law won’t cover care at Ronald Reagan UCLA Medical Center in Los Angeles, where he was referred under his current plan.

“I was forced to get it done this year,” said Mr. Rosenthal, who said waiting would have required him to get a new referral for another facility.