"So is there a maximum number of wires that can go through a single hole in a stud?"

Depends on the size of the hole.

Depends on the type cable.

Depends also on whether the plate is firecaulked and whether the wall is insulated.

Art 334.80 of the 2008 NEC, for example, requires that where more than 2 current carrying conductors in NM cables pass through the same opening in wood framing, and spacing is not maintained between the cables, and the opening will be firecaulked or sealed, the ampacity of the conductors must be derated.

The same derating of the ampacity holds true also for NM cables when installed in contact with thermal insulation.

It does not become so much a problem with how many cables can physically pass through the opening, but rather that the ampacity must be derated under the above conditions.

For 15A/14ga and 20A/12ga circuits you can go up to four **/2 cables or any combination of **/2 and **/3 cables that results in 9 or less current carrying conductors.

Derating starts from the full ampacity of the conductors, NOT the #14 @ 15 amp & #12 @ 20 amp values so the practical effect is not very limited.

The same applies to wires in high temperature areas like attics.

Sine NM now uses 90C (194F) conductors, that is were the derating starts, not the 60C (140F) limit for the cable assembly making this not very significant most of the time.

Complicated answer - it is a 3/4 in hole with 2 14/2 and 2 12/2s. Firecaulk and no insulation. So what I am hearing is that the wire is derated but not to a point that it would ever matter. But maybe I should drill a second hole....

Thank you so much for the answers. The NEC is truly impressive in its scope.

"Complicated answer - it is a 3/4 in hole with 2 14/2 and 2 12/2s. Firecaulk and no insulation. So what I am hearing is that the wire is derated but not to a point that it would ever matter. But maybe I should drill a second hole.... "

No, the condictors are not already derated.

And no, it is completely unecessary to drill additional holes.

Why?

Because even though the NEC says that you must derate these conductors because:

1. You have more than 2 current carrying conductors in the hole
2. Spacing is not maintained between the cables
3. The opening is firecaulked.

You begin to derate the conductor using its 90C ampacity for copper wire found in Table 310.16 and by the derating factor found in Table 310.15(B)(2)(a).

For THHN conductors, the 90C amapacity of 14 AWG CU conductors in a cable assembly is 25 amps.

12 AWG THHN CU conductors has an ampacity of 30 amps.

But when there are up to 8 current carrying conductors in close proximity, you need to derate the ampacities by 80%.

80% of 25 amps is 20 amps.
80% of 30 amps is 24 amps.

So even after derating from the 90C amapacity, each of these conductors can still carry full load currents.

Why?

Because the 14 AWG conductors will normally only carry 15 amps maximum...well below the 20 amps even after derating.

The 12 AWG conductors will normally only carry 20 amps, again well below the already derated value of 24 amps after derating.

Only time you would have problems with your 4 cables going through the same opening in the framing member is if you bundled 10 or more current carrying conductors into the same conduit, cable assembly or in close proximity to one another because the derating factor would be at least 50% and could be higher if you had to double-derate for ambient temperatures as well.

Bottom line:

Under normal ambient temperature conditions, your 4 cables through a 3/4" hole won't pose any problems even if firecaulked and insulated.

Do these same rules apply in an unfinished basement area and the floor joists?

In my home there is quite a number of wires running through the same holes particularly near the service panel.

manhattan42, that's a good explanation of HOW derating is to be applied to NM cable, but not WHEN it's to be applied.

You wrote above:
Art 334.80 of the 2008 NEC, for example, requires that where more than 2 current carrying conductors in NM cables pass through the same opening in wood framing, and spacing is not maintained between the cables, and the opening will be firecaulked or sealed, the ampacity of the conductors must be derated.

What 334.80 says is: Where more than two NM cables containing two or more current carrying conductors...

So derating is actually not required by 334.80 due to more than 2 ccc's in the hole, but rather due to the presence of a third cable in the hole.

"So derating is actually not required by 334.80 due to more than 2 ccc's in the hole, but rather due to the presence of a third cable in the hole."

Go back and read the first post in this thread.

The original poster said he has some spots where there are 3 cables in 1 hole.

Correction.

Previous post should read:

"The original poster said he has some spots where there are 3-4 cables in 1 hole."

And that is what he was concerned about.