Update your post to include full system hardware specs and OS information.
Is Hitman the only game failing to launch/start?
No pop-up windows, error codes, etc..?
Do you see any entries in Reliability History or Event Viewer that correspond to the times of the failures?
Make and model router? The router's logs (if available and enabled) may reveal some problem or reason.
Who has full admin rights to your router? You will need help from that person.