To answer some of your post, DHCP, the service that assigns IP addresses to your devices supports 2^8=256 IP addresses (minus a few reserved), so your ok with that. You want to try to isolate the problem.
Does connecting with ethernet solve this problem? If it does, sounds like a WiFi issue. If there are a lot of WiFi networks in your area, try changing the broadcast channel.
Can you connect to your LAN? Try to ping your router's IP address in command prompt. Then try to ping a website like google.
If your computer can connect to the LAN over WiFi and ethernet, but not to the WWW, an possible problem that I have encountered is a bad splitter. If you are using a splitter to divide your cable to your TV and router, try connecting the router directly to the coaxial instead of through the splitter.
Something that I have done is try specifying a different DNS server. My download speed is good but ISP has crappy DNS servers, so it does not resolve domain names to IP addresses quickly. You can specify your computer to query other DNS servers instead. Somewhere in your control panel, you can find your IPv4 settings and specify it to use Google's DNS address which is 18.104.22.168 and 22.214.171.124
You can just search "using google dns" and you can also configure your router to use these settings.
Remember, using the ping command is very helpful.
Hope that helps