Major insurers in some states are proposing hefty rate boosts for plans sold under the federal health law, setting the stage for an intense debate this summer over the law’s impact.

In New Mexico, market leader Health Care Service Corp. is asking for an average jump of 51.6% in premiums for 2016. The biggest insurer in Tennessee, BlueCross BlueShield of Tennessee, has requested an average 36.3% increase. In Maryland, market leader CareFirst BlueCross BlueShield wants to raise rates 30.4% across its products. Moda Health, the...