This paper explores the use of isomorphic problem pairs (IPPs) to assess introductory physics students' ability to solve and successfully transfer problem-solving knowledge from one context to another in mechanics. The paired problems are "isomorphic" because they require the same physics principle to solve them.  We analyze written responses, individual discussions for a range of isomorphic problems, and potential factors that may help or hinder transfer of problem-solving skills from one problem in a pair to the other. When quantitative and conceptual questions were paired and given back to back, students who answered both questions in the IPP often performed better on the conceptual questions than those who answered the corresponding conceptual questions only. Although students often took advantage of the quantitative counterpart to answer a conceptual question of an IPP correctly, when only given the conceptual question, students seldom tried to convert it into a quantitative question, solve it, and then reason about the solution conceptually. Even in individual interviews when students who were given only conceptual questions had difficulty and the interviewer explicitly encouraged them to convert the conceptual question into the corresponding quantitative problem by choosing appropriate variables, a majority of students were reluctant and preferred to guess the answer to the conceptual question based upon their gut feeling. Misconceptions associated with friction in some problems were so robust that pairing them with isomorphic problems not involving friction did not help students discern their underlying similarities. Alternatively, from the knowledge-in-pieces perspective, the activation of the knowledge resource related to friction was so strongly and automatically triggered by the context, which is outside the conscious control of the student, that students did not look for analogies with paired problems or other aids that may be present.