If you can set up a computer to act as an intermediary (i. e. a router between your computers/PlayStation/whatever and your gateway - Linux strongly recommended for this!), you can use ip(6)tables to log any traffic between your PlayStation and the 'Net (both outbound and inbound, especially any ICMP replies or returned TCP RST packets are of interest here) to find out some information like remote IP addresses and ports, then use tools like ping or traceroute (to check reachability, the latter can help pinpoint where the connection may be failing) and nmap (lets you determine the state of particular ports - use with caution! Especially make sure to check individual ports only as anything else is likely to be perceived as an intrusion attempt!).
If necessary you can also attempt to set up Wireshark to capture any network traffic between your PS and PSN and see what’s going on (again requires some IP addresses and/or port numbers to only capture the relevant packets - you should have determined them earlier with ip(6)tables).