Why obviously?

It doesn't mean "do you have an internet connection?" It means "are you currently, at this moment connected and using the internet?"

Have you read through the forums to find a problem that sounds like it is the same as yours? Have you tried all the solutions suggested to those problems?

If the problem is genuinely unique to you then you will need to provide a lot more information for anyone to try and diagnose it:

What operating system?
What version of Java?
What version of LimeWire?
What internet connection?
Firewall/router?
Any other software running?
Any other details that just might, possibly, perhaps have some bearing on why LimeWire won't run on your system?

Mark