There is evidence of recombination within a DENV serotype; see e.g.
Article Widespread Intra-Serotype Recombination in Natural Populatio...
and I hazily remember a report such a thing (interserotype recombination) appearing some time ago but of course now I can't find the paper and a quick online search turns up nothing. So my guess is, it should be possible, although if I had to guess I'd say it is much more likely to occur in infected mosquitoes.
there are few reports of concurrent infections in the same patient by two or more serotypes of dengue, and by molecular analysis of RNA via RT-PCR, recombination has not been demonstrated, although they probably merit deeper studies to be able to elucidate this with greater certainty
The problem is that for recombination to occur, a single cell has to be superinfected by a second virus (which is not the same as a single individual coinfected with two serotypes). I am aware of a few, not very detailed studies on DENV superinfection, but they are all in mosquito cells (C6/36) and they all suggest the existence of a strong interference with superinfection from a second serotype. Is anybody aware of studies in mammalian cell lines?