I'll try answering your question since it isn't specific to the Dis Boards:
In general, things just slow down until it can catch up with requests, or if it becomes worse, it could just stop completely.
Here's an example of how a server could slow down abnormally -
Consider the server to have 100 resources available.
Out of the 100, 14 have to do with just keeping the time of day, connectivity to the outside world, virus scan duties, and for record keeping duties.
Out of the 86 remaining, 16 are designated for 'high priority' duties.
Out of the 70 remaining, let's say they are all available for medium and low priority duties.
Let's say the server belongs to a store and they're having a sale, and it's a very good sale.
So if 100 people hear about the sale and try to connect to the server and browse the items for sale, it can handle the load by scheduling tasks and users, and things run normally, because it can take turns servicing each one, and some people will read the fine print.
Now 50 more people than the normal 100 come along and try to browse and buy things and we still only have 70 remaining regular duty resources, but now they have to take care of 150 people, or requests.
Once the server gets behind and has to stretch out the time to perform the requests, it kind of snowballs. Some server operating systems then reschedule their resources to try to adjust for the overload, which helps in some circumstances, but could hurt in others.
Sometimes, a small hardware problem could make the server not be able to complete requests, or to make it fail on only some of the requests.
So when the server becomes overloaded, it makes thing move in small segments, so it looks like it is stuck in goo and moving very slowly.
Hope this helps.