I started at a new employer last week, which uses the Cisco AnyConnect VPN client software. I installed it on my Windows 7 64-bit system. And the VPN miserably failed to connect, for no apparent reason.
Since I use a Cisco router at home, this was not likely to be the cause of the problem. I figured it might have something to do with the Kaspersky anti-virus that I use on it. So, I "paused" the Kaspersky protection, and tried the VPN connection again. And lo and behold, the VPN worked just fine.
A few google searches later, it appears that the workaround for this problem is to mark the Cisco vpnagent.exe executable as trusted in Kaspersky.
The procedure for KAV 2010 is described here.
The menus had slightly different names in KAV 2011 that I currently use, but were close enough. No reboot was needed to apply the settings. After this change, I was able to re-enable the KAV protection, and connect to the VPN.