Q: Safari can't open the page

I am SUPER frustrated with this issue!  All of the sudden, within the past 3 months, I have begun seeing this message for several sites I had visited regularly over the past years!  It seems to happen mostly with secure sites (https vs. http), and it's getting more prevalent as the days go by.  I type in or click from a bookmarked address, and get the "Failed to open page" screen, stating "Safari can't open the page because Safari can't establish a secure connection to the server "xxxxx.com".  I'm super angry because I do all my stuff online!

I have begrudgingly and regrettably reset Safari - ALL options! - and I still have this same issue with the same troubling sites (on top of losing all my saved information, top sites, etc.).  It seems each week, another 1 or 2 get added to this problem.  I contacted each company to let them know (via email), but I don't know what else to do.  All my systems are up to date as well.  I am running OS X 10.8.5 and Safari 6.2.8.  Anyone else have this frustrating issue??