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A hypothetical individual living in the US opened an HSA account in 2016. She was enrolled in a high deductible health care plan in 2016, 2017, and 2019.

She kept records of all her qualified medical bills totaling to $4000 from 2016 and 2017 but she did not need to tap into her HSA to pay for health care services.

Can she withdraw $4000 in 2019 from her HSA without penalties?

Yes.

From HSABank:

Can I use my tax-free HSA savings to pay for — or reimburse myself for — IRS-qualified medical expenses from a previous year?

Yes, as long as the IRS-qualified medical expenses were incurred after your HSA was established, you can pay them or reimburse yourself with HSA funds at any time. Just be sure to keep sufficient records to show that these expenses were not previously paid for by another source or taken as an itemized deduction in any prior tax year.

IRS Pup 969 confirms this, but not as succinctly:

For HSA purposes, expenses incurred before you establish your HSA aren’t qualified medical expenses. State law determines when an HSA is established. An HSA that is funded by amounts rolled over from an Archer MSA or another HSA is established on the date the prior account was established.

The accepted answer is great, I just wanted to point out that this feature is why some people make maxing out HSA contributions a higher priority than IRA's. With an HSA you can get pre-tax contributions, tax-free growth, and tax-free disbursement if used on qualified medical expenses. The longer you wait to get reimbursed for qualified expenses the more tax-free growth you can get.

Even if you don't want to sit on receipts for years, contributing to an HSA and not using it immediately is still useful in retirement. Once you hit 65 you can use the HSA to pay for Medicare parts A, B, D and Medicare HMO premiums. Also at 65 you can take penalty-free distributions for any reason, but you would pay income tax on those distributions (like a traditional IRA). Over-funding an HSA isn't much of a concern.

These rules could change in the future, but currently the HSA offers fantastic tax advantages. One important caveat, is that not all HSA's have good investment options, so evaluate your plans investment options before making HSA a priority.