  We investigate interlacing properties of zeros of Laguerre polynomials $
L_{n}^{(\alpha)}(x)$ and $ L_{n+1}^{(\alpha +k)}(x),$ $ \alpha > -1, $ where $
n \in \mathbb{N}$ and $ k \in {\{ 1,2 }\}$. We prove that, in general, the
zeros of these polynomials interlace partially and not fully. The sharp
$t-$interval within which the zeros of two equal degree Laguerre polynomials $
L_n^{(\alpha)}(x)$ and $ L_n^{(\alpha +t)}(x)$ are interlacing for every $n \in
\mathbb{N}$ and each $ \alpha > -1$ is $ 0 < t \leq 2,$ \cite{DrMu2}, and the
sharp $t-$interval within which the zeros of two consecutive degree Laguerre
polynomials $ L_n^{(\alpha)}(x)$ and $ L_{n-1}^{(\alpha +t)}(x)$ are
interlacing for every $n \in \mathbb{N}$ and each $ \alpha > -1$ is $ 0 \leq t
\leq 2,$ \cite{DrMu1}. We derive conditions on $n \in \mathbb{N}$ and $\alpha,$
$ \alpha > -1$ that determine the partial or full interlacing of the zeros of $
L_n^{(\alpha)}(x)$ and the zeros of $ L_n^{(\alpha + 2 + k)}(x),$ $ k \in {\{
1,2 }\}$. We also prove that partial interlacing holds between the zeros of $
L_n^{(\alpha)}(x)$ and $ L_{n-1}^{(\alpha + 2 +k )}(x)$ when $ k \in {\{ 1,2
}\},$ $n \in \mathbb{N}$ and $ \alpha > -1$. Numerical illustrations of
interlacing and its breakdown are provided.
