This is a list of common issues people don't seem to realize with the PSP's connection. If you have issues connecting your PSP to an Access Point, run through these in order.

1. SSID encryption support: None, WEP, or WPA (TKIP or AES)
People often make the mistake thinking that WPA2 is supported by the PSP due to the PSP displaying "WPA", however it only supports the initial release of WPA. Some routers will actually blast a signal that accepts both WPA and WPA-2 encryption commonly known as "mixed WPA" but is probably not in use if your connection is WPA2 (CCMP) or WPA2/PSK.

2. Wireless interface support: 2.4 GHz 802.11b+g (Wi-Fi 3)
It might support 2.4 GHz 802.11b+g+n (Wi-Fi 4) but I have not tested that. It does NOT support 802.11g+n (Wi-Fi 4) which most access points probably use at this point. It does not support anything that uses the 5 GHz band.

3. "Not Encrypted", "No Encryption", "None" Security options may not be what you think.
Some routers however not all! often mask these options as a passwordless WPA2/PSK (or whatever is relevant) encrypted network. As it's still encrypted, your old device won't be able to connect to it, regardless of the fact that there's no password.

4. DHCP support
While this is more of a "have it in the back of your head" sort of p.s., its to let you know that old devices like this tend to have issues with DHCP systems on modern routers. However, I don't specifically know why. If you think this could be the case of your connection issues, all I can recommend is attempting to specify a value to connect with rather than using "Auto" for IP Address, Subnet Mask, and Default Gateway (Also known as Router IP, and Default Router IP). The values you should use is different for everyone, your router's administration portal will be the best bet on telling you what values your network configuration can accept. If you can't find anything, try to connect on another device and see if it tells you the values it ended up using in a Test Connection or View Information/Details pages, this may provide hints on the values to use. Do not copy-paste the value for IP Address, as the value must not be already in use when trying to connect with a new device, instead change the last value of the IP address to something that isn't in use (do not use "1" as that will be reserved to the router, and typically the first few values are the oldest connected devices).

All information shown unless stated alongside it has been tested and verified with a FRITZ!Box 7530.
If you have any suggestions on stuff to check, please do let me know, and ill sticky your post.

 

[86houdini]

Protip: Most routers have the ability to broadcast a separate band of Wi-fi with different settings. One thing you can do, and something I do personally, is to set up an open network for older devices like this, or a DS Lite. If you're worried about security, you can hide the SSID, or use MAC address filtering.