Thinkstock Images/Stockbyte/Getty Images. Generally, people become eligible for Medicare coverage at the age of 65. However, there are two circumstances in which you can get coverage at 62, or even sooner. In some cases, coverage will begin automatically.n the other hand, if your benefits started at age 61, you aren't Medicare-eligible until you turn 63. People who have been diagnosed with Amyotrophic Lateral Sclerosis, or Lou Gehrig's disease, are eligible for Medicare in the same month they get disability benefits.