So, in theory, if I connected to a network using, say, 30 ethernet cables from the same PC, wouldn't the upload speeds go up exponentially, because it would split the data over all the ethernet connections and send them to the router separately (sorry if this is absolutely bizarre. I'm in no way a network engineer and know nothing about network engineering)
In most instances for hosts, no. Each interface is generally a separate network.
Some hosts (usually servers) can use multiple (generally a maximum of eight) connections bonded together as a single virtual link, but there are some ways in which this could actually degrade performance.
If you divide all the packets in a single network flow across multiple links, you can increase lost packets and packets delivered out of order. This can cause performance problems, and it may cause some things, e.g. real-time applications like VoIP, serious problems.
If you assign a single flow to a single link, and you have multiple flows, then you can see increased bandwidth overall, but each flow will still be limited to the speed of a single link.
Depending on the network switch vendor, this is called Link Aggregation (LAG) or etherchannel. This is commonly used between switches, or from servers to switches. This is controlled by algorithms which decide which flow gets assigned to which link. The algorithms generally work best if the number of links is a power of two, e.g. two, four, or eight.
You also must think about what happens later in the network. If your host has four 1 Gb links to the switch, but the switch has a single 1 Gb link toward your destination, then you have oversubscribed your bandwidth, and you can lose a lot of packets because the switch is dropping three out of four frames you are sending.