Given the variability in how airlines deal with pets and carry-on pets, I'd heartily recommend everyone traveling with a pet or two check the airline's policies before purchasing a ticket - and at least prior to travelling. (I'd even print the policy out and carry a hard copy in case of any dispute with an airline employee.)

Northwest's Carry-on Pet policy is easily available at its website, and in part it states:

N.B. Referencing the OP, it also states: For the safety and comfort of your pet, please make sure your pet can fit comfortably inside your carry-on kennel. The kennel must be large enough to allow your pet to move about. No part of your pet may extend outside of the kennel. This includes long tails on some birds. (It does not state they must be able to stand up and turn around... either.)

Accomodation: Your pet must always remain in its carry-on under your seat for the duration of the flight. Northwest limits each flight to six (6) carry-on kennels; two (2) in First or WorldBusiness Class and four (4) in Economy. On Northwest Airlink flights operated by Mesaba and Pinnacle Airlines, there is a limit of four (4) carry-on kennels: two (2) in First Class and two (2) in Economy.

Delta's pet policy states: Your pet must remain inside the kennel (with door secured) while in a Delta boarding area (during boarding and deplaning), a Delta airport lounge, and while onboard the aircraft.

RSSrsvp has kindly established a "sticky" FAQ thread (at the top of the Forum) with links to many airlines' pet policies. It's there to be used, for your convenience.