Is there a limit to the amount of items a pipe can carry at once, or, a limit on the number of items within a chunk?

I've setup my own Tekkit 3.1.3 server which I'm playing on with a couple of friends. We setup 3 quarries in fairly close proximity. Each quarry's output is piped to a main transport pipe which pipes a long distance back to our main hub. Everything is running fine except that we keep finding random piles of overflow items by the pipe. These aren't as a result of a chest being full. Rather, they are along the length of the actual pipe. The problem disappears if we only run one quarry at a time.

So is there a limit to the number of items the pipes can transport at a time or is this something to do with an item limit per chunk? If so, is there a way to increase this limit?

No, there's no limit to how many items can be in a chunk, nor in a pipe. I've piped a rotary macerator at full speed into a cobblestone pipe, and it handles them fine. The limitations of pipes are designed entirely around transport speed, not contents capacity. Minecraft's limitations on items aren't designed into the game mechanics — rather, too many items will simply overload memory and slow your server to a crawl.

Since you have a long pipeline, you are likely having parts of it unload when there are no players around that section. When this happens, the items can drop out of the last section of loaded pipes since they aren't "connected" to anything. You're probably seeing this more with all the quarries running simply because more items mean more chances to have part of a pipe that should be receiving items be unloaded at any given time. The easiest way to get around this is to make sure the chunks that have pipes running through them are always loaded by using World Anchors, Anchor Carts, or another block that designates a chunk to stay loaded even with no players around.

(As an alternative to buildcraft pipes you could also use RedPower2 pneumatic tubes. These won't transport items through unloaded chunks, but they have a more favourable failure mode: when a pipe can't deliver, items just rebound and get "backed up" in the tubes until there's somewhere for them to go. This would lower your throughput compared to a proper Anchor solution though, and would require rebuilding your item transport system over the entire long distance.)