The problem is, since TSA was created, that was never true, nor is it today.On both US domestic and international scheduled commercial flights, you are allowed no more than one carry-on and one personal item. The airlines haven't, nor do they expect to in the future, permit a third carry-on.
I know a few photographers who mistakenly believed that TSA page about photo equipment, which was thankfully taken down 2010. They were forced to check one of their carry-ons, by their airline. Some made the mistake of checking their camera bag, and when they retrieved it, found some or all of their equipment was missing or damaged.
Unfortunately, nothing seems to ever go away on the Internet, and cached copies of the defunct TSA “camera bag page” still show up from time to time, and catch unwary travelers. Don't let it be you.
At the gate, the airline's gate agent checking you through to board your flight will likely count the number of your carry-ons, and eyeball their size and weight. You might get away with an extra carry-on, but usually not. You might get away with a carry-on which is a little large and/or heavy, but recognize you're taking a chance. If you happen to get a gate agent who's having a bad day, watch out! Your oversized roller carry-on will then likely be refused entry to your airplane's cabin.
Suggestion: To give you more room in your carry-on and personal item, carry your camera/lens over your shoulder, or hang it from your neck (no super telephotos though) I've never seen any airline not permit that in addition to one's carry-on and personal item.When I turn in my boarding pass and walk into a plane, my camera/lens (wide angle zoom, typically) is hanging from my neck. I'm wearing my photographer's vest to carry odds and ends. I'm wearing my photo equipment backpack. Admittedly, it's not exactly light, and its size may be somewhat of a stretch as a personal item, but as I'm wearing it, it's not yet been questioned. I also wheel in my roller carry-on. Its physical size conforms to every domestic and international airline standard. It even fits “endwise” into the old 737 overhead bins, which are about the smallest flying these days.
(Please note: Due to the small overhead bin, and overall cabin size, typically everyone must “gate check” roller carry-ons, when flying on regional jets and propeller driven aircraft.)
Let's talk about tripods.
There is no doubt, that system-wide, TSA, and for that matter, their counterparts in airports throughout the world, have not made a definitive determination as to the safety of tripods, as, or in, carry-ons, on today's scheduled commercial airline flights.
TSA agents make their decision about whether or not passengers may bring a tripod into an airplane's cabin on a minute to minute, completely ad hoc basis. Therefore, I only have rules of thumb to help photographers bring tripods with them when flying.
If I have checked luggage, I pack my tripod in my checked luggage, sans head.The head is in my carry-on. I carry-on the head because it's far more difficult to replace if its lost, stolen or damaged. To date, I've never had a tripod damaged or stolen, when packed in my checked luggage.
With regard to TSA, packing your tripod in your checked luggage is the safest way to bring it with you.
Believe it or not, TSA TSO's (Transportation Security Officers) have enormous flexibility in deciding what's allowed on airplanes and what's not. It doesn't have to be on the prohibited list to be refused.
Last September and October, for example, I saw a tripods in the outside pocket of a photo equipment backpack just peeking out, and one neatly strapped to the outside of a camera bag, refused to be allowed through TSA security. TSA TSO's required the two tripods be checked.
In the last year or so, I've not seen TSA refuse to permit tripods packed totally inside carry-on bags, sans spiked feet, but I have seen them refused in the past, and there is no telling what TSA might do in the future.
This I know for sure. If a traveler attempts to bring tripod as, on, or in carry-on, with spiked feet, it will be refused.Beyond that, from my personal experience with TSA, and through interviews with them, if your tripod is wholly inside your carry-on, without spiked feet, it will most likely be permitted, but I can't say absolutely, positively. That's far too definite for TSA.
Experientially, I can guesstimate that you have a 50–50 chance, at best, getting your tripod into your airplane's cabin if you hand carry, or tie it to the outside of your carry-on. More than a few TSA agents consider tripods brought into airplane cabins, that way, potential weapons.
You've also have to get your tripod past your airline's gate agent. I've seen many gate agents refuse to allow tripods which were hand carried, even though TSA let them through. They force it to be at least gate checked and frankly, it's not likely to survive undamaged if gate checked.
I've see gate agents regularly refuse tripods, tied to the outside of carry-ons because they explain it violates the airlines linear size restriction for carry-ons. Frankly if it sticks past the top of the bag, it likely doesn't meet their linear size limitation.
Here's my strong suggestion. If you're taking a tripod, put it in your checked luggage, if any. For travelers with only carry-on bags, pack your tripod completely inside your carry-on and leave its spiked feet at home.