Get your taxes done using TurboTax

Yes, you would qualify for the Short Gap Exemption.  Since you had insurance on May 31, you are counted as having insurance in May.  Since you had insurance through Feb 17th, you are counted as having insurance for February.  The months you were without coverage would be March and April.  This qualifies as a lapse of coverage of less than 3 months. If your insurance did not start until June 1st, then no, this would not apply. 

• In any tax year, you may apply for a Short-Gap exemption if you had a lapse in coverage of less than three months.
• New Jersey considers you covered during a month if you had coverage for even one day during that month. So if you are without insurance for two consecutive months, you must have insurance by the last day of the third month to qualify for a Short-Gap exemption.
• If you have a gap that is three months or longer, you cannot claim this exemption for any month. The Shared Responsibility Payment will be due for every month you were without coverage.
• If you had two or more such gaps in coverage during a year, you can claim this exemption only for the months of the first coverage gap. Example: You didn't have coverage any day in May or any day in November or December. You can claim the exemption only for May.
• If your coverage gap crosses calendar years, the months without coverage of the second tax year aren't counted for the exemption for the first tax year. But the uncovered months from the first year are counted for the exemption for the second tax year. Example: You don't have qualifying coverage in November 2020, December 2020, and January 2021. You're not eligible for the 2021 short gap exemption for January because you didn't have coverage for three consecutive months - from November 2020 through January 2021.

If you qualify, you can claim this exemption for the dependent(s) you claim on your tax return.  NJ Shared Responsibility Requirement

If you did not start insurance until June 1st, you may also qualify for a hardship exemption.

As you walk through your NJ return in TurboTax, you will be given the opportunity to fill out the exemption for to have the penalty waived.