  We study interlacing properties of the zeros of two types of linear
combinations of Laguerre polynomials with different parameters, namely
$R_n=L_n^{\alpha}+aL_{n}^{\alpha'}$ and $S_n=L_n^{\alpha}+bL_{n-1}^{\alpha'}$.
Proofs and numerical counterexamples are given in situations where the zeros of
$R_n$, and $S_n$, respectively, interlace (or do not in general) with the zeros
of $L_k^{\alpha}$, $L_k^{\alpha'}$, $k=n$ or $n-1$. The results we prove hold
for continuous, as well as integral, shifts of the parameter $\alpha$.
