No, the search problem is not solved. When I look at the referrer logs of my site, I find people coming from as high as the 100th search result page of Google. My experience with searching on the Internet also leads me to believe that search needs to get better and better. A lot of very interesting things are happening in the search field; vertical, social, etc..
But, I will have to agreed that the major Search Engines are extremely good at the present moment. I use Yahoos Search API on my site, and am happy to fiddle around with their API's rather than build my own crawler (although I have real people adding links).