How to find the cause of high ping times to the Internet

Cable modem-->Cisco ASA5510-->Netgear 48 port switch.  
All workstations running Windows XP, with the exception of one laptop with Vista.  We are experiencing a slow Internet connection with ping times varying from 100ms-400ms.  Ping times to workstations and servers within our network are normal.   Ping times early in the morning before staff arrives, normal.  Ping times return to normal in the evening when only a couple of work stations are still active.
The following is what has been done so far to try and find the problem:
Reset cable modem, Cisco ASA5510 and Netgear switch and all subsequent switches in our network.  No change.  
Connected directly to cable modem, ping times to Internet are normal.
Experienced high ping times while directly connected to Cisco ASA5510.  If Cisco is the culprit, why normal ping times before everyone arrives and after everyone leaves?
Monitored ping times while unplugging each workstation from Netgear switch.  No change.
Today found one possibly malfunctioning NIC card on a server.  Switched to another card, no change.
Swapped out Netgear switch, no change.
Will try bringing one workstation at a time online in the morning.
We are a non-porfit organization, so resources are slim.  It has been a couple of years since I used Wireshark, so I will require some quidance with it.

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How many users on this network?  Are they all doing a fair amount of work on the internet?
charismatic100Author Commented:
25 workstations.  Most of the work is done internally.  There are times when work is done on the internet, files uploaded/downloaded.  At first, thought that it was from doing MS updates on laptops for deployment coupled with payroll processing.  Times would return to normal with each update that was completed then go back up when starting a new one.  This anomally started last week and has gotten progressively worse, to the point where once the times go up they remain there.
When I left last night, there were two employees left and times were normal.  When I got here this morning, ping times high again.  Found one workstation locked when I had requested everyone to shutdown before leaving.  Unplugged from network, reset cable modem, netgear switch and Cisco ASA5510.  Ping times normal.
Sounds pretty obvious that the Cisco is the culprit.  The question is "why?".  You said that once the ping times go up they don't come back down again until you either reset the router or everyone leaves and the traffic drops to near zero.  Perhaps the Cisco is overheating.  The more traffic it sees the more the CPU has to work, causing internal temps to rise.  If it has insufficient cooling then it stands to reason that once it gets hot it can't cool back down until the CPU can quit working so hard.  I have a Netgear router at my house that is like this.  During the summer it required rebooting twice a day on average.  I replaced the router, but I still have the old one because it works perfect until it heats up.  So I keep it for short diagnostic purposes.

If you have access to a router that you could perhaps borrow for a day or two.  Swap it with the Cisco and see what happens.  If that solves your problem perhaps you can hold some sort of fundraiser for the money to get a new router.
charismatic100Author Commented:
Good point, which is often overlooked.  But overheating issue was suspected and then ruled out.  
I replaced a NIC card on one of the workstations after receiving a complaint about slow access time to data server and Internet.  At the same time,  my supervisor changed ports on the switch for the server, which we had done previosly with no results.  We had completed both tasks when he reported that ping times to the Internet had returned to normal.  They have been normal ever since.

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