The '53 Yankees hold the record with a 9 game losing streak during the season while winning the WS. The 2006 Cardinals had two separate 8 game losing streaks and won the WS. The '05 White Sox had a 7 game losing streak. Not sure how many have had 5 game losing streaks but I bet it's a fair amount.
Of course, the longer this goes on not only the worse our chances get but it also solidifies that we just aren't very good.
It's not quite panic mode yet, particularly with the 3 wild cards. 88 wins or so likely gets us in and we aren't that far off from that pace. Teams that are 25-23 are legit contenders. Teams with far worse records to start have gone on to win the WS all the time.
But if we drop the next 3 games to the Rays we will be 12.5 back of them and I think the chances of winning the division is probably gone. WC will still be in play though.