Driving from Cape Cod to JFK for an evening flight - feasible?

If you really want to avoid the worst of the Connecticut and NYC traffic, or if you want to break up your trip in the middle, you could always use the Cross-Sound Ferry from New London, CT to Orient Point, NY (on Long Island.) The voyage takes about an hour and a half, and costs $55 (one-way) for a car & driver, $15 for each additional adult, and $6 for each additional child. Departures are at least once per hour during daylight hours, and sometimes twice per hour.

This route doesn't actually save you any driving time (Google Maps shows 2 hours from Barnstable to New London and 2 hours from Orient Point to JFK); and instead of CT traffic, you'll have to deal with inbound traffic on the Long Island Expressway instead. (Choose your poison.) But it might be a fun thing for the kids to do; if nothing else, it'll give you a break in the middle of the day to get some food and let the kids stretch their legs.