Size-based scheduling strategies such as Shortest Remaining Processing Time first (SRPT) and Least Attained Service first (LAS) provide popular mechanisms for improving the overall delay performance by favoring smaller service requests over larger ones. The performance gains from these disciplines have been thoroughly investigated for single-server systems, and have also been experimentally demonstrated in web servers for example. In the present paper we explore the fundamental stability properties of size-based scheduling strategies in multi-resource systems, such as bandwidth-sharing networks, where users require service from several shared resources simultaneously. In particular, we establish the exact stability conditions for the SRPT and LAS disciplines in various limiting regimes. The results indicate that size-based scheduling strategies may fail to use the available resources efficiently, and in fact cause instability effects, even at arbitrarily low traffic loads, and will therefore not yield optimal delay performance. The qualitative findings confirm the tendency for users with long routes and large service requirements to experience severe performance degradation.