Uncoordinated multiple access has been proposed as a modern technique to provide massive connectivity. Among many proposals, irregular repetition slotted ALOHA (IRSA) was shown to achieve a close-to-optimum throughput by irregular packet repetition and successive interference cancellation (SIC). However, most of the previous works are based on certain ideal assumptions. In this paper, we tackle one of such ideal assumptions, where a decoded packet can always be used to cancel out all its replicas transmitted in other slots. In reality, the random nature of IRSA results in a large dynamic range of the received power that cannot be recovered under a practical quantizer. A SIC limit that dictates the number of recoverable collided packets at a slot should exist. To resolve this issue, we propose a novel protocol, named IRSA with frame partitioning (IRSA-FP), which partitions a frame into several subframes. A feedback mechanism is enforced to inform those users, whose packets have been successfully decoded, to cease their transmissions. We then propose a novel density-evolution-type analysis for IRSA-FP. Simulation results show the performance of the proposed IRSA-FP can be accurately predicted by our analysis and the proposed IRSA-FP outperforms the traditional IRSA when practical SIC limit is introduced.