Are you logging in with Active Directory accounts?

There are a bunch of things to try but I thought I'd ask this first!

Yeah, we're using Active Directory accounts.

When I get this issue it's normally a broken connection to AD. I test it using dscl in interactive mode.

When you were testing from the terminal, what commands where you trying?

Also, do you get any error messages at the login window or does it just shake?

My coworker was the one that said he tested from terminal. He just used the id command with a username to see if it'd find it. I've asked to have the teachers notify me the next time a computer does it so I can get in and do more extensive testing.

No error messages, it just shakes you off.

Ok, id can be misleading as it will just give you cached results, even if you have mobile accounts switched off. To use dscl to test, open a terminal window and do the following:

dscl
cd Active Directory/YOURDOMAINNAME/All Domains/Users - (I would use tab to auto-complete to avoid typos on the above command)
ls

You should get a list of users back from AD. If you get errors leading up to this it probably means the connection to AD is breaking.

A common cause is the machines set to go fully to sleep. try setting a computers energy saver settings to put the display to sleep, but not the computer to sleep, and uncheck "Put hard disks to sleep when possible". It may be that the Mac is going completely to sleep and cutting the network connection, then failing to reconnect to AD when it wakes up).

Thanks! I'll test this as soon as I get another reported computer. Shouldn't take too long...

I turned off sleep on a small set of the computers so I can watch the behavior. You already inadvertently fixed the other problem I was having on the computers, that they weren't following their startup/shutdown schedules. Turns out that is because of sleep...they weren't waking up to shutdown at night. So thank you!! Hopefully this turns out to the be the cause of the AD issues as well.

Good to hear! Hopefully that will fix the AD issue as well.

The binding issue finally appears to have been fixed in 10.10 :)
With the waking up at night, if you do want to be able to use sleep... I have a policy that runs pmset to scheduel and wake at whatever time I need to perform maintenance, works pretty well.
I scope it to classroom machines that have updates pending and it wakes them at a quasi random time just after 1am the following morning. It's using wakeorpoweron so it actually powers on an off machine as well, but you can just use sleep if you just want to shut them down.

#!/bin/bash
TomorrowsDate=`date -v +1d +"%m/%d/%Y"`
RandomMinutes=`date +"%S"`
pmset schedule wakeorpoweron "$TomorrowsDate 01:$RandomMinutes:00"

Thanks for all the help on this. Its definitely an issue with sleep in that the computers aren't waking up. This was screwing up both logging on as well as the startup/shutdown schedules. I'm not finding much out on the web about this being a Mavericks bug or not.

From the sites I've worked with its an issue thats been around for most versions of OS X. We see the same problem where users leave network drives connected and the Macs go to sleep, particularly if they have files open from those drives. They find that they can't save them when they wake the Mac back up again.

You can use configuration profiles in Casper or the pmset command to set the necessary energy saver settings.