This phase I trial is studying the side effects and best dose of temsirolimus when given together with lenalidomide in treating patients with previously treated multiple myeloma. Lenalidomide may stop the growth of multiple myeloma by blocking blood flow to the cancer. Drugs used in chemotherapy, such as temsirolimus, work in different ways to stop the growth of cancer cells, either by killing the cells or by stopping them from dividing. Temsirolimus may also stop the growth of cancer cells by blocking some of the enzymes needed for their growth. Giving lenalidomide together with temsirolimus may kill more cancer cells.