How is that possible? I distinctly remember trying some new application that was faster than what I used before and I couldn't understand how it could be so fast. Does anyone remember this happening? Unfortunately I don't remember what software it was exactly.
But if speed limited not by hardware but by software means - firewall, p2p could download much faster, if you fortunate enough to download something popular.
And from my own experience near decade working in telecom domain, they rare limited by hardware, mostly they have much more hardware speed than offer you, because they usually limited on trans-ocean connections. So they limit speed for clients by software, so trans-ocean connection divided by clients equally. If not limit client connections, faster machines will got much faster download, I have seen 3 times difference on machines in one room, connected to one network switch, and without speed limits.
And these software limits are usually not constant, because network usage is highly changes during time of day, limits also change, accepting you to download faster when little load.
For example, my connection once had hardware limit 100mbit ethernet, and outside network of my provider, they guarantee 20mbit (I don't know current limits, for many reasons, most reason war).
When download with http/ftp/ssh-copy, I usually see something like 2-3mBytes/s at business day peak time, and 3-4mbytes at night, but when try hot popular movie (I choose with most number of downloads on tracker), I usually see 7-10mBytes/s - close to theoretical max of 100mbit ethernet if one way usage (when also upload, could be 100+100mbit=200mbit).
It blew my mind every single time.
This kind of behavior of caching can be seen in programs like mpv [1]; with streaming works wonders.
(1) Might you have had an ISDN modem and two 56kbps channels? Or are you certain it was a plain 56kbps modem and an ordinary POTS line (plain old telephone service).
(2) Are you sure that the P2P software wasn't lying to you? Did you time it yourself? It might have been showing you a highly exaggerated transfer speed (maybe basing the speed on the uncompressed file).