The main aim of this paper is to provide cosmological constraints on the Multi Scalar Field Dark Matter model (MSFDM), in which we assume the dark matter is made up of different ultra-light scalar fields. As a first approximation, we consider they are real and do not interact with each other. We study the equations for both the background and perturbations for N -fields and present the evolution of the density parameters, the mass power spectrum and the CMB spectrum. In particular, we focus on two scalar fields with several combinations for the potentials, however the work, along with the code, can be easily extended to more fields. We use the data from BAO, Big Bang Nucleosynthesis, Lyman-α forest and Supernovae to find constraints on the sampling parameters for the cases of a single field and double field, along with the Bayesian evidence. We found that some combinations of the potentials get penalized through the evidence, however for others there is a preference as good as for the cold dark matter.