As you're probably already aware dynamic IPs are allocated at random and reused when required.
As a result tracking of dynamic IPs by a copyright holders is notoriously unreliable due primarily to issues such as IP reallocation, proxies, and the potentially large number of hops between the end user and the Internet backbone.
I'd guess that the copyright holder has misidentified your IP and although you weren't active your router was presumably connected to this misidentified IP address.
Again I'm only guessing but I'd imagine that all the ISP has done is warn the user of the IP specified by the copyright holder and I'd doubt that they've bothered to check to confirm if a large download was actually carried out using that IP at the time alleged.
Tell them again that you weren't downloading anything at that particular time and demand that they provide all the evidence they have that the IP address in question was being used for large downloads at the time they allege.
With luck, and providing there isn't anything more sinister going on here, it should be fairly obvious that the connection wasn't being actively used.
Does your router have any logs that might support any of the above or confirm which IP you were connected to at the time in question?