There are three things you might want to do using relativity: (1) describe an object that's accelerating in flat spacetime; (2) adopt a frame of reference, in flat spacetime, that's accelerating; (3) describe curved spacetime. General relativity is only needed for #3.
A prohibition on #1 is particularly silly. It would make SR into a trivial theory incapable of describing interactions. If you believed this, you would have to stop believing, for example, in the special-relativistic description of the Compton effect and fine structure in hydrogen; these phenomena would have to be described by some as yet undiscovered theory of quantum gravity.
Number 1 often comes up in discussions of the twin paradox. A good way to see that general relativity is totally unnecessary for understanding the twin paradox is to pose a version in which the four-vector equation a=b+c represents the unaccelerated twin's world-line a and the accelerated twin's world-line consisting of displacements b and c. The accelerated twin is subjected to (theoretically) infinite accelerations at the vertices of the triangle. The triangle inequality for flat spacetime is reversed compared to the one in flat Euclidean space, so proper time |a| is greater than proper time |b|+|c|.
Number 2, accelerated frames, is less trivial. It's for historical reasons that you'll see statements that SR can't handle accelerated frames. Einstein published special relativity in 1905, general relativity in 1915. During that ten-year period in between, nobody really knew what the boundaries of applicability of special relativity were. This uncertainty made its way into textbooks and lectures, and because of the conservative nature of education, some students are still hearing, a century later, incorrect assertions about it. There is an overwhelming consensus among modern relativists that the boundary between SR and GR should be defined as the distinction between flat and curved spacetime, not unaccelerated and accelerated observers.[MTW 1973,Penrose 2004,Taylor 1992,Schutz 2009,Hobson 2005]
In an accelerating frame, the equivalence principle tells us that measurements will come out the same as if there were a gravitational field. But if the spacetime is flat, describing it in an accelerating frame doesn't make it curved. (Curvature is invariant under any smooth coordinate transformation.) Thus relativity allows us to have gravitational fields in flat space --- but only for certain special configurations like uniform fields. SR is capable of operating just fine in this context. For example, Chung et al. did a high-precision test of SR in 2009 using a matter interferometer in a vertical plane, specifically in order to test whether there was any violation of Lorentz invariance in a uniform gravitational field. Their experiment is interpreted purely as a test of SR, not GR.
MTW 1973 -- Misner, Thorne, and Wheeler, Gravitation, 1973, p. 163: "Accelerated motion and accelerated observers can be analyzed using special relativity." p. 164: "An accelerated observer can carry clocks and measuring rods with him, and he can use them to set up a reference frame (coordinate system) in his neighborhood."
Penrose, The Road to Reality, 2004, p. 422, "It used to be frequently argued that it would be necessary to pass to Einstein's general relativity in order to handle acceleration, but this is completely wrong. [...] We are working in special relativity provided that [the] metric is the flat metric of Minkowski Geometry M."
Taylor and Wheeler, Spacetime Physics, 1992, p. 132: "DO WE NEED GENERAL RELATIVITY? NO! [...] 'Don't you need general relativity to analyze events in accelerated reference frames?' 'Oh yes, general relativity can describe events in the accelerated frame,' we reply, 'but so can special relativity if we take it in easy steps!'"
Schutz, A First Course in General Relativity, 2009. Schutz equivocates on pp. 3 and 141 about the status of accelerated observers in SR, but says, "[...] the real physical distinction between these two theories is that special relativity (SR) is capable of describing physics only in the absence of gravitational fields, while general relativity (GR) extends SR to describe gravitation itself."
Hobson, General Relativity: An Introduction for Physicists, 2005, sec. 1.14, discusses "Event horizons in special relativity" from the point of view of accelerated observers, using coordinates defined in their accelerated reference frames.
Chung -- http://arxiv.org/abs/0905.1929