Same issue, bought the Pebble Time watch last weekend as a gift for my wife, it worked fine with my phone Galaxy S2 for a few hours, then we paired her phablet Samsung Galaxy A 8.0 and soon after she got her first disconnect as seen on the watchface and on settings screen on the watch. The android app, however, did not detect disconnection and kept showing that the watch is connected. It keeps happening now and then every day. The only solution so far is to trigger airplane mode since disabling and enabling BT on her phablet does not help.
EDIT: also have not used Stand-By at all.