That, quite frankly, is bullsh-t. Having the money to pay for a sleeper doesn't make you a superman that can somehow magically carry more bags than the poor peons back in coach. Yes, you get more - a private room, a bed, meals. You don't get to just break the rules, no matter how special it may make you feel being able to pay that extra bit of cash.
Airlines are a poor example, because the limits on baggage are so low.
Ryan is right.
Unfortunately too many things are getting conflated in this discussion. So let us split things up logically and discuss it. I will use United as an example airline since it covers the world, and is not terribly unsimilar to most other US based airlines. There are some well known exceptions though.
1. Checked baggage: United does give extra free allowance on checked baggage, both in weight and number of pieces, based on class of travel booked and elite status. For example, because of my Platinum Elite status I am allowed upto three checked bags of upto 70lb each free of charge irrespective of class of travel.
But note, that this is not the subject of this thread.
For someone traveling using a low fare, with no status and no Explorer or equivalent or better affinity credit card, you get to pay even for the first checked bag.
2. Carry-on bags: AFAICT the same two bag rule applies to everyone irrespective of class of travel or elite status. You are allowed to book an additional seat to carry an extra bulky piece on board occupying a seat apparently, but I have never seen anyone actually do that. I suppose that might be a way to transport your Cello safely.
Note that this is the subject of this thread.
Also note that other than booking an additional seat no excess carry-on baggage is allowed. So there is no question of having an excess carry-on charge other than of course booking an additional seat.
3. Gate check: More often than not you don't get charged for a gate checked bag, but technically you could be if it is beyond your free checked bag allowance. You never get charged for a gate checked bag on a Regional flight AFAICT.
A small comment about "breaking the rules", I think it is a legitimate thing to discuss what the rules should be, without being blamed of "breaking the rules". However, if the rules are in place such as they are, I agree you should not get to arbitrarily break them just because you think you are special for whatever reason.
I am sure there are other models used by other airlines and it might we worthwhile bringing those up if they are relevant to this discussion. I would be the last one to claim that what United does is ideal in any way shape or for. It is just an illustrative example.