  We study bond percolation of $N$ non-interacting Gaussian polymers of $\ell$
segments on a 2D square lattice of size $L$ with reflecting boundaries. Through
simulations, we find the fraction of configurations displaying {\em no}
connected cluster which span from one edge to the opposite edge. From this
fraction, we define a critical segment density $\rho_{c}^L(\ell)$ and the
associated critical fraction of occupied bonds $p_{c}^L(\ell)$, so that they
can be identified as the percolation threshold in the $L \to \infty$ limit.
Whereas $p_{c}^L(\ell)$ is found to decrease monotonically with $\ell$ for a
wide range of polymer lengths, $\rho_{c}^L(\ell)$ is non-monotonic. We give
physical arguments for this intriguing behavior in terms of the competing
effects of multiple bond occupancies and polymerization.
