When opening Task Manager to investigate the issue, there was plenty of physical memory available, however the Commit (GB) was almost at it's limit. (see below)
One process that we noticed running high Working Set (Memory) and Committed Memory was svchost.exe.
The svchost.exe process covers a lot of services, such as AeLookupSvc, BITS, CertPropSvc, gpsvc, IKEEXT, iphlpsvc, LanmanServer, MMCSS, ProfSvc, Schedule, SENS, SessionEnv, ShellHWDetection, Themes, Winmgmt, wuauserv.
As in our case, it is extremely likely the service in question is wuauserv (Windows Update Service). This service scans the system and checks the Windows Update source to see if there are any updates to be downloaded and installed. This can use a lot of memory! If you temporarily stop the wuauserv service (on the services tab in Task Manager), you will see the svchost.exe process's memory drop drastically. Of course, keeping this service in a stopped state is not a viable long term option as the system will not receive updates.
There are two hotfixes available to address this:
1. High memory usage by the Svchost.exe process after you install Windows Management Framework 3.0 on a Windows-based computer https://support.microsoft.com/en-us/kb/2889748
Assume that you have a computer that is running Windows 7 Service Pack 1 (SP1), Windows Server 2008 R2 SP1, or Windows Server 2008 Service Pack 2 (SP2). You install Windows Management Framework 3.0 by installing update 2506143 or update 2506146 on the computer. In this situation, after the computer runs for a while, you experience high memory usage by the Svchost.exe process.
2. Windows Update Client for Windows 7: June 2015 https://support.microsoft.com/en-us/kb/3050265
This update addresses an issue in which system performance can be decreased during scans. This issue has the greatest effect on computers that have a small amount of physical memory.
Implementing these hotfixes improved our situation a lot.
If you still experience issues, you may need to review the amount of physical and virtual memory assigned to that system.