Once you had coverage other than an HDHP (and Medicare is an example of this), then you are no longer eligible to contribute to an HSA.
Actually, the calculation is on an annual basis, so you actually should base your 2020 HSA contribution limit on either 9 or 10 months out of 12 (9 months if you began Medicare on October 1, 2020 and 10 months if you began Medicare after October 1, 2020).
So you could contribute to your HSA for 2020, but you will not have the full annual HSA contribution limit. But in this way, it doesn't really matter in which months you actually made the contributions.
I am sure that if you indicated which months you had Medicare, that TurboTax did these calculations for you.
You added "My HDHP did not end due to my disability". Are you referring to the question at the end of the HSA interview where it asks what type of HDHP coverage did you have on December 1, 2019, and then asks if your coverage ended because of a disability?
If this is what you are referring to, come back and say so, and I will explain what this is referring to.
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I too have the same question about the plan coverage lapse.
My wife had a kidney transplant which is covered under the ESRD Medicare, and they pay for her immunosuppressant drugs for three years, but that is all they pay for. She never had a lapse in her HDHP coverage and still has it.
How should that situation be entered?