This started happening a few weeks ago, when I accelerate from a stop or slow speed, the engine kind of stutters; almost feels like it's going to stall. It started a few weeks ago, then went away for a couple of weeks, and now it's started again. Any ideas?

 

These engines are extremely prone to carbon fouling if you don't use Top Tier gas exclusively. We've seen severe fouling in as little as 10K miles.

You didn't mention how many miles on your vehicle. There is an extended warranty to 150K for 2015 and certain 2016 models (link below) for the fuel injector / rail system. I had mine replaced at 88K under warranty.

If your car hasn't thrown any codes yet, p0420 in particular is one to watch for.

https://static.nhtsa.gov/odi/tsbs/20...08198-0001.pdf

 

My money is on trans issue unless the firing issue is present during idle or other rpm ranges. CVTs suck.

 

The Fit CVT's are extremely reliable. OP hasn't given his mileage but I'd look at spark plugs, coils, fuel injectors.

 

I'm at around 172k miles.

 

Did you have it replaced before any problems showed up?

 

I seem to have partially identified the cause. So it happened in mid-January for a week or so, and then went away. Again, it started last week, then it seemed to go away by yesterday. I looked at my credit card statement and BOTH times, immediately before it started happening, I filled up at a particular gas station (supposedly Top Tier) near my home. Those were the only two times in the past 2-3 months that I went to that station.

Both times, the issue went away after I refueled at different stations and drove my car for a gallon or two.

 

You can get water into the gas station's storage tanks, especially if drainage is poor on their lot. You can also get unscrupulous station owners stocking lower quality gas than what they advertise.