We give a complete answer to the question of (semi)stability of tangent bundles on any nonsingular complex projective toric variety with Picard number 2 by using combinatorial criterion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundles on all toric Fano 4-folds with Picard number \leq 3 which are classified by Batyrev [1]. We construct a collection of equivariant indecomposable rank 2 vector bundles on Bott towers and pseudo-symmetric toric Fano varieties. Furthermore, in case of Bott towers, we show the existence of an equivariant stable rank 2 vector bundle with certain Chern classes with respect to a suitable polarization.