Compressive sensing (CS) has been very useful for the Internet of Things (IoT). CS aims to reduce the number of samples acquired and transmitted by a sensor node using a low complex sampling operation. Furthermore, distributed compressive sensing (DCS) was introduced where the compressively sampled sensors' readings are jointly recovered at the fusion center rather than being recovered separately as in the CS. As a result, a reduction in the number of required measurements as well as the complexity of the sensor nodes is achieved. None of the previous works had studied the performance of DCS for sensed signals with abnormalities caused by sensor malfunctioning, sudden changes in temperature, or humidity. (i.e., practical IoT networks). In this paper, we extensively study the performance of DCS when signals with abnormalities are considered using simulations. The results show that DCS outperforms CS in successfully recovering signals with abnormalities.