Yes, although as is pointed out at numerous points in the comments this is because of effects of motion through air. Take away the air and you would have an admittedly idealised experiment, but one in which indeed the dropped bullet and the fired one would take the same amount of time to drop. Gravity doesn't act any differently on a moving object than it does on a stationary one, but if it's in motion then other forces may also be in play is all.
The other problem is of course that weight and mass get mixed up all the time. Naturally we calibrated our mass scale on earth, where mass and weight can be expressed in essentially the same units -- on the Moon a 100kg man would only "weigh" 17kg or so, while the same person on Jupiter would weight something in the region of 240kg, before he was squished into goo.
When you are accelerating your weight doesn't actually increase, ever, but over forces can act on you in the same direction to make an "effective" weight. Again, the force of Gravity isn't changing, but once in motion other forces can come into play and either act against or with gravity, or in a direction that has no effect on your weight.