Beyond reasonable doubt is the standard at a criminal trial. To conduct a search, you only need probable cause, which is a lesser standard.
But if it is not reasonable to believe that an IP address is connected with a person, it is not justifiable to search or seize anything else that person may own or possess, beyond the particular device identified by the IP. I don't think an IP address alone is sufficient to even meet the probable cause standard.
It is also possible that someone with authorized access to his computer used an unsecured browser to view a compromised advertisement on an ordinary website, which recruited the machine into a botnet, which used infected machines as a distributed filesystem.
It is also possible that a wardriver with a cantenna brute-forced his Wi-Fi WPS PIN and extracted the WPA password, obtaining easy access to the network.
It is also possible that the ISP regularly refreshes its IP leases, and gave the cops identifying information for the customer who had the IP at the time the request was made, rather than the customer that had the IP at the time the suspicious traffic was logged.
None of those scenarios are necessarily exculpatory alibis, but they may cast doubt on any hypothesis that purports to connect an IP to a particular person.
In practice, warrant-signing judges seem to believe that an IP address is analogous to a postal address, possibly because it also contains the word "address", which may be conceptually ingrained into their minds as a permanent physical location.