Is it possible to file an extension on April 16th in Massachusetts?

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Yes!  Residents of Maine and Massachusetts have until April 17, 2019 to file a federal extension.  How do I file an IRS extension (Form 4868) in TurboTax Online?

Massachusetts individual extension

If you filed your Massachusetts individual extension Form (M-4768) prior to the deadline but were rejected, you have until 10 days after the filing deadline (April 27, 2019) to submit another extension.

You can re-file here.