The Cardinals are 4-11 in the month of June. They have 9 games remaining this month and have to go 8-1 just to be .500 for the month of June. They have 3 @ Nationals, 2 vs Cubs in London, 3 vs Astros and 1 vs Yankees. Will the Cardinals be able to go 8-1 versus those teams? Maybe they get 5 wins versus the Cubs and Nationals but that means they can only lose one of the final 4 against the Astros and Yankees. That is asking a lot.
In comparison the Brewers already have double the wins the Cardinals have in June. The Reds have a recent winning streak that is double the Cardinals win total in June. Are the Cardinals out of it? No they are not out of it yet but if they don't start winning and soon they will be out of it. For the Cardinals to catch the Reds, Brewers and Pirates they would have to hope for those teams to lose the rest of their games.
The Brewers have 11 games left in June, Reds 10, and Pirates 11. So far the Brewers are 8-8 in the month of June. If they continue to play like they have they probably finish 5-6 or 4-7 in their final 11 games in June. The Reds if they continue playing how they have been playing probably go 7-3 or 6-4. Then you have the Pirates who are not really playing all that well either lets say they match the Brewers at 5-6 or 4-7. If we take the worst for each that is 6-4, 4-7 and 4-7. The Reds would be up by about 2 games at the end of June. If the Cardinals go 8-1 they only pick up two games and are still 8 games back. They are currently 10 games back as of today (6/18).
Going into July the Cardinals would have to go 22-4 or something like that to be back in the race for the NL Central. I just don't see this as probable for the Cardinals.