Yes, the contributions made to your Health Savings Account (HSA) by your insurance company are counted toward your total contribution limit. Therefore, the maximum that you can contribute would be $5400 by your example.
According to the IRS Publication 969, 'you can claim contributions you made and contributions made by any other person, other than your employer, on your behalf, as an adjustment to income.' So, be sure to enter the contribution into your tax return. You will include it as if you personally made the contribution.
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