They yesterday looked to be cruising to the points and took a 2-0 lead back to their dressing room at half time.
But in-form East Thurrock had other ideas. They stormed back to earn and 2-2 draw and could have won the match.
Bluebirds manager Collier says there is no hiding place after winning promotion last season and that focus must be greater at a higher level.
"We really should have been out of sight at half time, 2-0 up and creating a lot of chances," said the Chippenham manager.
"It could have been 3 or 4, but we didn't get the second or the third goal and they got back into it at 2-1.
"Before we knew anything they got level and that came through a lack of concentration. At this level of football you get punished. We have switched off and you are then looking to save a point."
Collier added: "If you give people too much time and too much space, unfortunately these things happen.
"Last season (in the Southern League) maybe we would have got away with that but not in the National League South. The teams and the players are too good.
"Thankfully now we have a free week before our next came so we can regroup and go again."