Usually not: if you are covered by a general-purpose health reimbursement arrangement (HRA) that can pay your medical expenses before you meet your deductible, you generally can’t contribute to a health savings account (HSA), even with a high-deductible health plan (HDHP).
IRS Publication 969 lists several kinds of HRA that do let you keep contributing to an HSA alongside an HDHP. A limited-purpose HRA pays only for items such as dental care, vision care and preventive care. A post-deductible HRA pays nothing until you have met the HSA minimum annual deductible. A suspended HRA is one you elect to pause before the coverage period starts, and a retiree-only HRA pays only for expenses incurred after you retire. Limited-purpose and post-deductible health FSAs work the same way. HRAs are funded only by employer contributions, never by employee salary reductions, and unused HRA money can carry forward to later years. Because the HRA design decides your HSA eligibility, check your employer’s plan documents before you set up HSA contributions.
Source: IRS Publication 969, Health Savings Accounts and Other Tax-Favored Health Plans
Image: Stethoscope on a piggy bank by 401(k) 2012, CC0, via Wikimedia Commons.