Assuming you're confident the problem isn't low propane/gas supply, the first thing I'd check is for spider webs (or some other kind of blockage) in the burner tubes.
This is actually a fairly common problem; enough so that small brushes intended to address this exact issue have been on the market for a pretty long time. You also may be able to address the problem by blowing out the obstruction using can of compressed air.
Check your grill's documentation for info as to the location of the combustion vents on your particular Weber.