For the trip discussed here, I'm considering flying with my wife from JFK to Venice, arriving on a Sunday at about 9am. We would leave Venice about 5:30, me on Tuifly to HAJ, she by train to Bassano. So we would get to fly over together, plus we could have lunch in Venice.

But: is there a place to store our luggage at the airport? Are there any other potential gotcha's? Is this a goofy thing to do?

The alternative is that I fly directly to HAJ from New York, and she flies by herself to VCE. Would be ok, but not as much fun.

Gotchas? How about being really, really tired?

Also, unless you pay for a water taxi - which only takes about 25 minutes to get to the city but costs 100EU - you will spend most of your time traveling from the airport to the city and back.

Actually, I think that for the right kind of person, this is the perfect romantic, crazy, fun thing to do!
IF you have everything planned out ahead of time, that is. I'd suggest hiring a private water taxi driver for the day. They'd pick you up at the airport around 10am after you'd cleared customs and baggage claim, and take you directly into the city. You would just have time for lunch and a little bit of wandering around. Your boat would always be close by, at your disposal, and would store your bags for you. Then the driver would take you back to the train station and airport.

Not cheap, but stress-free and smooth.

Without the private driver, there are a whole lot more what if's and maybe's, as well as a great deal of work on your part (what to do with bags, getting from airport to central Venice, finding your way around, finding your way back, etc. It's doable, but requires more work on your part.)

I'm confused about storing the bags at the airport. I would think you could either check your bags through or recheck them upon your arrival at VCE. And, if your wife is leaving by train, her bags would be at the wrong place.

If you have the money to burn on something really romantic, hiring a water taxi for the day sounds like a very luxurious way to go!

and this would be the only feasible way to do it.

And if you do it, don't wait until the day of and pick a driver from the dock - the price will be unbelievably loose and high!
Set it up before hand so that the price is firm.

You probably have your plans made by now, but just FYI, there is a left-luggage office at Venice Airport. Also, a bus into Venice takes around 20 minutes and the Alilaguna shuttle takes around an hour (water bus). I think its perfectly do-able, unless you are delayed....