Deaths increase on Mt. McKinley after rescue policy initiated (Clark and Lee paper here)
Accidents increase in NASCAR with adoption of additional safety (my paper with Russ Sobel here)
The examples of off-setting behavior are abundant. Auto insurance and reckless driving, theft insurance and locking doors, government bailouts and risk-taking by firms, etc., etc. When people receive some sort of protection against an unfavorable event they exert less effort trying to avoid that event, ultimately increasing the probability of that event occurring. Incentive matter.