We consider the first-crossing-time problem through a constant boundary
for a Wiener process perturbed by random jumps driven by a counting process. 
On the base of a sample-path analysis of the jump-diffusion process
we obtain explicit lower bounds for the first-crossing-time density 
and for the first-crossing-time distribution function. In the case of the 
distribution function, the bound is improved by use of processes comparison 
based on the usual stochastic order. The special case of constant jumps 
driven by a Poisson process is thoroughly discussed.