Can GWT email me if a broken link is found on my website?
Google Webmaster will only email you if it finds critical issues with your site. For example, it will email you if it detect malware on your site or see a significant increase in errors while crawling your site.
They recently announced this via their blog: http://googlewebmastercentral.blogspot.com/2012/10/keeping-you-informed-of-critical.html
You can change some of those notifications by going to your email notification preferences. https://www.google.com/webmasters/tools/preferences