For federal tax purposes, no.  A charity can be totally supported by fee for service income and still qualify under section 501(c)(3), so long as you don’t carry on any substantial unrelated business activity.  You should have no problem qualifying as a public charity under section 509(a)(2) because of the large number of relatively small fees from your patients.  (See Ready Reference Page:  “Calculating Public Support Percentage”)

Whether you will qualify for state charitable exemption from real estate or sales taxes will depend on the state in which you are located.  Some states require a “subsidy” in order to qualify for state exemption.  They require you to show that the revenue earned is less than the cost of the services provided. (See Ready Reference Page: “Act 55 Defines ‘Charity’ Eligible for [PA] Exemption”)  You may qualify by having the higher fees on the scale cover a portion of the cost of the lower fees, but it is a lot less difficult to prove the subsidy if you are using current grants and contributions to make up the required subsidy portion of your services.

The prime benefit of federal (c)(3) status, of course, is to allow you to give deductibility to individual and corporate contributions and to qualify for foundation grants.  You might as well use that status to get money to extend your program, and if you are in a state that cares, show the subsidy necessary to qualify for state exemption -- and extend your program farther.