If you want something else, you have to ask. They can say yes or no, depending on other traffic and factors.
If you depart with less than full length they will have to say so on the air so it is recorded on the tapes for any possible accident investigation if you do it wrong.
Ronachamp can fix this if it's not correct phraseology, but say you've asked to depart at Alpha 14 in this diagram below...
"Cessna Seven Niner Mike, cleared for takeoff, intersection departure, Alpha 14." is the minimum they're required to say. I've also heard some controllers add "X feet remaining" where X is the number of feet left on that runway for takeoff from that intersection. I assume that's a courtesy or a big hint that it's SHORT from a little placard somewhere in front of them in the tower cab. (Plus, they'll have it memorized.)
It's NOT meant to be confirmation that what you've requested is a good idea for your aircraft or even possible, only that it meets THEIR requirements and they don't think you're completely nuts.
It's still on you as PIC to know there's adequate runway for takeoff and also knowing where you are going to go if the engine quits now that you threw away a few thousand feet of runway.
Which of course also changes your takeoff briefing to yourself and/or the whole crew if you're PIC but have other pilots on board, and your TOLD (Takeoff and Landing Data) you previously worked out.
I typically won't accept an intersection departure at an unfamiliar airport. It's an extra card stacked against you if the engine quits. If the intersection chops off only a small fraction of the runway, okay. Not thousands of feet. Better to have that runway ahead as an option for an engine out.
However, I have accepted and done the math for the Alpha 14 at KAPA when it's cold out.
Two reasons, usually I'm doing it to get out ahead of a jet that's taxiing ahead of me down to go full-length, and I know there is a golf course/driving range directly off the north end of 35R. I have at least one option, sucky as it may be.
At the end of the day though, the NTSB report will include that I accepted the risk and lost, if I end up in the golf range... And I will be angry at myself for accepting he Alpha 14 departure. Rationalization won't change the additional risk incurred.
I will not ask for Alpha 14 with passengers. It's my Jon to lower the risks for them and they have no idea that I'm adding risk.
Flip side... I will accept and ask for Runway 10 if winds are light. It's slightly riskier than 35R because it's downhill and there's a berm off the end you have to clear to get to an open field.
Lots of decisions, us pilots. Just sharing my thought process for you. You get to make your own!
Cool, isn't it?