Physically seen, there might be a perfect signal, but there has to be a logical connection also. If that lacks, there's no Internet, or only 'limited connectivity'.

You can see it in the wireless icon in the 'system notification area'. If there's a logical connection, it has some circle segments in the upper right. And you can see it in the properties. A sign of malfunction also is that there's no IP-address mentioned in the report of IPCONFIG (run from the command prompt).

Common causes include:
- wrong encryption type or wrong password defined for connection in XP
- MAC address of wireless card excluded (or: not included if it set to filter) in router
- the 'wireless autoconfiguration service' in XP is disabled
- router not set to issue a IP-address (DHCP functionality)
- connection in XP not set to ask for an IP-address

Some remarks:
- tells more. It might need some time to sort it out and configure it correctly.
- It's really an exception if this is caused by defective hardware, especially if the signal is OK.
- More often than not, it's on XP side, not on the router side.
- Generally, XP SP2 does it all without need for additional software, but some wireless cards still come with software for pre-SP2 systems that you can't disable. That complicates things, because the standard XP functions are modified. To get it running on my daughters laptop, with WiFi built in, I found it easiest (after struggling for some time) to uninstall Fujitsu's supporting software. On my own older laptop, Belkin's management software comes automatically with the driver, disables XP functions, and can't be removed without removing the driver. But it works. Don't know about Linksys. My desktop, luckily, is wired.

Good luck with your endeavours.