Memory and Disk Space both affect your internet connection.
Memory is important to a speedy internet connection because your computer tries to load its software into Memory (RAM). Without sufficient RAM, your computer will use your hard drive as an extension of its system memory. The Rule of Thumb is: The more Memory (RAM) the better.
Disk Space: If your system hard drive (usually c:\) is full, you can expect to see diminished performance from your computer and internet connection. You will likely see slower performance under the following conditions:
If your computer's hard drive is heavily fragmented, it will cause system performance problems due to read/write operations having to take longer. Under severely fragmented conditions, the slowdown will be quite noticable. Anything over 20% is considered severe.