Can I purchase Supplemental Health Insurance while covered by a HDHP and HSA?

[HSAEnthusiast]

Can I purchase supplemental and/or secondary health insurance while being covered by a HDHP and contributing to an HSA? I was thinking about purchasing a secondary and/or supplemental health insurance plan so I wouldn't have to dip into my invested HSA funds if I became sick, hospitalized, or was injured. I was thinking about purchasing some kind of Gap policy, an accidental medical expense policy, or some other type of supplemental health insurance plan to help cover my deductible. Can I do this while enrolled in a high deductible health plan and maxing out an HSA?

[Bsch4477 · Accepted solution]

You can’t have the usual medi-gap insurance. Certain disease specific plans are allowed. See page five of this publication. 
https://www.irs.gov/pub/irs-pdf/p969.pdf

[Opus 17]

You generally can't have any "other coverage."  The whole point of the combination of an HSA and HDHP is that if you have a high deductible plan (which policymakers like for various reasons, but leaves you with unpredictable expenses) you are allowed this powerful tax-free savings tool to save up for those unpredictable expenses.  Getting supplementary coverage to make your expenses more predictable, defeats the purpose of the HDHP with HSA.

As noted, there are a few specific exceptions, see the linked instructions.