I'm a new homeowner and pretty inexperienced with gardening. At the
beginning of Spring when my lawn was overrun with dandelions and
weeds, I ran out to Kmart and bought a 50lb bag of weed n feed. I
stupidly didn't read the instructions figuring I couldn't do much hard
because it was a "feed". Well, I was definitely wrong. I "spread" it
with a cup and dumped a whole cup on areas of large weeds. Well,
needless to say, and much to my horror, have many large patches of
brown dead grass!!! It just looks horrible!

At this point, is there anything at all I can do aside from digging up
the many dead patches and re-seeding? Is there any chance that the
grass will grow there again?

What were you thinking using a chemical product without reading the label?

You've burned the grass out. You need some good, hard rains to wash this
further into the soil (or a few weeks of heavy sprinkler work). In the
spots where you "dumped a whole cup" you may want to take off the top couple
of inches of soil, throw it out and replace it.

After the fertilizer has been thoroughly washed down (probably 4 or 5 HEAVY
and prolonged rains or a few weeks of thorough sprinkler action every other
day) go out and buy several bags of compost to spread about the property.
Spread it thinly -- we're not looking to build an inch layer here, just add
to the soil a bit. Then get some Scot's Sun & Shade mix (or another quality
grass) and overseed the lawn. Water regularly and the grass will appear.

You need to mow this new grass regularly -- as in every time it gets to
about 4 inches in height. Keep it at about 3 inches. Do not let it grow
more than 4 inches before you mow -- you'll stress the plant and may kill
it.

Another option is to simply resod the yard or to hire a contractor to come
out and handle the reseed.

And FWIW -- stay away from chemicals, you rarely truly need any such thing.
They're really no good for the soil or plants, certainly no good for you and
your family, and all you really do is ensure that you'll need to keep
applying chemicals. Each year, spread some compost on your lawn, mow
regularly, and pull weeds by hand -- you'll be happy you did.

James