Non-orthogonal multiple access is a promising technology for the fifth generation systems which exploits the power domain to achieve higher spectral efficiency. The performance of NOMA techniques are usually investigated under an ideal setup with perfect successive interference cancellation. However, the limitations of NOMA techniques under a setup with imperfect successive interference cancellation are not well understood. Contrary to the approaches in the literature, we examine the performance of NOMA under a non-ideal setup and propose two power allocation algorithms. The first algorithm is designed for the max-min problem whereas the second algorithm considers the heterogeneous rate requirements of users and provides solutions based on a novel rate measure. The performance of the algorithms is investigated both theoretically and numerically under a non-ideal setup with channel estimation errors. The theoretical analyses reveal that the algorithms achieve the optimum power allocation for the rate max-min problems. The numerical analyses are not only in agreement with the theoretical analyses, but also show the superiority of the proposed algorithms compared to both the conventional multiple access techniques as well as other NOMA approaches.