Name of ISP: Time Warner
Country of Residence: USA (New York)
Server: Gilgamesh
When does the problem occur: Last night is the most recent occurrence at 11:00PM EST to around 2:00AM EST. It has happened other nights around typical primetime hours, so around 8:00PM EST to about 1:00AM EST.

Game stops entirely for upwards of 10 seconds every 30 seconds or so. Last night it was during a coil run and my group was able to see my own actions as I was running around a stationary boss and party. I was even able to do some mechanics as they were called out despite nothing happening on my screen.
Any assistance would be appreciated.