It's possible to do that, but you really have to work hard and take special care to do it right, or you can face penalties and interest on the not-quite-avoided payroll taxes.
Remember, authorities at every level (IRS, SAIF, Employment Dept.) have a very serious interest in making sure that only those people who are truly independent contractors get treated that way. And don't think that the "contractor" wanting to be treated that way cuts any ice with anybody. People have a funny way of having memory lapses in times of trouble -- such as when budget cuts force you to drop the contractor, who only then realizes that "independent contractors" can't claim unemployment, or when they fall down the stairs while working for you and only then learn that they don't have any worker's comp coverage (of course, not having worker's comp also might mean that they could have a right of action against your agency for personal injury).
If you're only going to know one thing about this area of law it's this: Anyone who does work for pay is an employee unless you establish otherwise, and the burden is on you, the employer to prove otherwise; in other words, if you are using an independent contractor in your nonprofit and someone decides that you have misclassified that person and that you should have been paying payroll taxes on their wages all along, it is you who has to prove them wrong. (If you fight them to a tie on the issue, then you lose, because you failed to carry your burden.)
The opinion announced today in Oregon is a well-written, clear cut guide to the issue that anyone who is using independent contractors might want to review and discuss with an attorney versed in this area.