I have a DSi, and I've had it for over a year now, with no trouble at all. I've treated it with the utmost care and it has responded very well.
Recently, as in the last two days, I haven't been able to get my DSi connected to the Internet anymore. I don't know what happened in the mean time, but suddenly I just can't. I can search for an access point, and the nearby routers appear. I click on mine, and test the connection, but the error code 052001 comes up, saying that it cannot obtain an IP address.
After that, I manually input the computer's IP address (adding 10 to the last number, so that instead of 188.8.131.52, it is 184.108.40.206), Subnet mask, and Gateway, along with the DNS and secondary DNS. I test the connection then, and the error code 052101 comes up, saying that it is unable to connect to the interent, and to check the connection settings and access point settings. I tried this at my girlfriend's house and the exact same thing happens. So it's not my router. Mine has WEP encryption, and my gf's has WPA. I used to be able to surf the Internet even if it was WPA, but now I just can't get online. I tried formatting the system, but nothing has improved.
Have you tried updating your DSi?