First, a definition: "Synaesthesia is a curious condition in which an otherwise normal person experiences sensations in one modality when a second modality is stimulated."  There are two basic explanations for this phenomenon: the hyperconnectivity
hypothesis and the disinhibition-unmasking hypothesis . The former says that the condition results from excessive neural connections between two modalities, and the latter says that it results from a lack of inhibition of signals. A quick search reveals that it is indeed possible to induce this condition (with drugs, for example), and it isn't necessary to rewire any neural connections [2,3,4]. This is an example of how the brain transmits/filters information by many mechanisms, connectivity being just one of them.
 Ramachandran, Vilayanur S., and Edward M. Hubbard. "Synaesthesia--a window into perception, thought and language." Journal of consciousness studies 8.12 (2001): 3-34.
 Kadosh, Roi Cohen, et al. "Induced cross-modal synaesthetic experience without abnormal neuronal connections." Psychological Science 20.2 (2009): 258-265.
 Ramachandran, Vilayanur S., and Diane Rogers-Ramachandran. "Synaesthesia in phantom limbs induced with mirrors." Proceedings of the Royal Society of London. Series B: Biological Sciences 263.1369 (1996): 377-386.
 Luke, David P., and Devin B. Terhune. "The induction of synaesthesia with chemical agents: a systematic review." Frontiers in psychology 4 (2013).