There is no problem with Google Webmaster tool. However it happen when Google update its data according to its new algorithm. I hope it will be temporarily and you will get back your backlinks. I think you have set sitelinks in webmaster tool and due to this it is seeing in tool. It means is not that your sites will show sitelinks in actual search result as well. Whenever Google will feel that your sites' sitelinks are useful for user then it show in search results.
What is the relation between backlinks and sitelinks in GWT ?
There is none. I observed that sitelinks are generated from the most visited pages from the homepage. Ex. A visitor arrives at your site homepage and clicks on services page. If this happens most of the time when your site receives a homepage level traffic, your services page will most likely be returned as one of the sitelinks the next time your site homepage rank for a keyword.
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