Cisco VPN connects but cannot ping and there are no network resources available.

What is most troublesome about this problem is that the behavior has occurred on several of my machines (laptop and otherwise).  The pattern usually goes something like this:
1.  I install the VPN software on the machine and connect just fine.
2.  Network resources are available and life is good.
3.  I disconnect and don't connect for a few days.
4.  The next time I connect it is successful but, no network services are available and I am unable to ping a networked machine.

As stated earlier, I have tried this on several machines and sometimes I am able to connect several times before the networking resources are barred from me.  I use a direct connection to a cable modem (no router).  The cisco VPN version is 3.6.3.  Does anyone else find this a bit strange?