sure, you can run tunnels across tunnels. though the end result at the PC may be different depending on which one you initiate first. as well, some VPN clients lock you out of all local network access during its connect. and then, some VPNs may not even let you out on the internet to even connect to the WWW (you may not be able to connect to your privacy VPN after the work VPN is connected. ) It just depends on the policies the particular VPN is set up with.
you can dump the PCs routing table to see the difference after each connect (netstat -nr). run 4 tests and check the routing table after each one. 2 test just singly connecting VPN by themselves, the 2 more tests running both (3rd test connect 1 VPN first, the the other, 4th test, swap order)
then, some VPN clients periodically adjust the PC routing table in case you try to manually put a route in to circumvent VPN policy.
lots of vpn solutions out there some even include packet filters that attach to your network card and they filter traffic from there per whatever policy they include