A quick google shows that there is actually a version of traceroute that does what you want, using TCP-packets.
No idea if something similar exists for Windows.
If you run Linux, try to use that tool. It could give you an indication where the packets are blocked. If you don't run Linux, maybe you have a laptop with Linux lying around somewhere.
This free utility could work too.
But it will only tell you: works or doesn't work. It won't tell you which router is blocking the packets.
The same guy also has a tcp traceroute for windows.
Best solution is always: quick phonecall to your network admin. There might be a reason stuff is blocked. If it's intentional or not, you probably need his help anyway to get your db-access working.