334.80 Ampacity (2017)
Says that the ampacity of NM cable shall be determined by 310.15, and the allowable ampacity shall not exceed that of a 60 degree C rated conductor.
For NM cable, 14 gauge is rated at 15 amps at 60 degree C.
At 90 degrees C, it is rated at 25 amps.
More than 10 current carrying conductors means a 50% ampacity adjustment.
50% of 25 amps is 12.5 amps.
12.5 amps is less than the 15 amp 60 degree C rating for 14 gauge, so it is not compliant.
Less than 10 conductors is adjusted at 70% or less, which comes out to 17.5 amps with 14 gauge wire, which would be compliant.
You will find that 12 gauge romex maths out the same way, under ten conductors is at 70% adjustment, and it works, above ten is 50% adjustment, does not work.
So therefore, more than 4 pieces of romex being bundled together for more than 24" will be a violation, because 4 romex = minimum 8 current carrying conductors, which would be adjusted at 70% = compliant. 5 romex would be a minimum of 10 current carrying conductors, so adjusted at 50% = not compliant.
If I'm wrong, please correct me - I wrote this up this morning and on Friday and I would need to eat crow while its warm...