In a word (or two) It Can. BUT, there are a few things to consider.
First you have to actually eat what you plant. One year I planted a TON of beets. While we like beets, maybe not a tons worth. Alot of them went bad and some went to the animals or straight to compost. Another year I planted a whole bed of radish's and then forgot about them (dont ask) by the time I noticed them, they were literally the size of my arm and a little tough (actually alot tough).
Another thing to consider is planting realisticly. Last year I wanted fresh corn for the winter. I planted 32 stalks figuring we can eat some fresh and then freeze the rest to have fresh corn in winter. It took up alot of square footage and considering the fact that corn takes a long time to grow I could have gotten 2 crops of a faster growing vegetable and had more food. In the end, with 1 cob per stalk, I had a whopping 3 cups of corn to take me through winter. Lesson learned, I will leave the corn growing to someone else.
Seeds can be a little pricey also, but there are ways around that too. You can attend "seed swaps" and swap what you have and try something different that you probably wouldn't have bought normally. You can also save your own seeds, which requires a little know how depending on the seed, but pretty easy to do.
There is also the time aspect. Gardening will take time. Planning, starting seeds indoors, planting outside, weeding, remembering to water, weeding, checking for bugs, weeding, fertilizing, weeding, harvesting and more weeding.
With all that said gardening is good for the soul, gets you out in the fresh air and if you get your kids involved (to help with the weeding) you'll have conversations you would not normally have.
So, start your garden and even if you dont save a penny, its OK, you'll still be eating healthier and (hopefully) having fun doing it.