Saying no doesn't have to be the end of the line for you, and it doesn't have to imply disrespect. What it can mean is that you're overwhelmed, and you need your manager's help prioritising your work. One of a manager's fundamental responsibilities is to help workers sift through what's important to work on now versus what can be worked on later. (In my experience, often it wasn't the work with the noisiest sponsors that was the most urgent.)
No one's suggesting that the next time your boss walks up and asks you to take on a new project that you outright say "I'm too busy". Instead, it's more helpful to point out to your manager exactly how heavy your current workload is. Let them know that if you take on new work, something you're managing now could possibly fail. Ask for their help deciding which task is more important so you can fit it in to your workload.
It's not as clear as just saying "Nope, sorry," but it gets the same message across: you have enough on your plate, and if you add something else, you'll need their help deciding what comes off. That way even if you do wind up having to reluctantly say yes, you'll hopefully get the room to do the job right without being overwhelmed, or at least help sorting priorities.
If you manage your own workload, it can be even easier. Chris Brogan takes the idea a step further and suggests responding with a clear "Thank you for thinking of me. I'm going to have to pass," and follow up with a referral to another colleague who can and would appreciate the reference. He says he'd rather say no when approached with work he can't perform than apologise when the project failed because he couldn't devote the right amount of attention to it.
In the end, it's more important to be negotiable but clear that you have enough current priorities that you shouldn't take on anything more. Remember, saying no tactfully doesn't mean you'll always get your way, but it can mean that you have more say in what work does or doesn't land in your inbox.
How to Say No [ChrisBrogan.com]