No, it is never possible for a set of n vectors to span R^m where n is less than m. R^m has a dimension of m, so m vectors are required to span the space. In the case of three vectors in R^4, the three vectors span a space with a dimension of 3 and R^4 has a dimension of 4, so vectors can be found in R^4 which are outside the space of any three vectors.
