It works to get one pull, but it's harder to get 'em all pushed though. Personally I think 4BM is overkill and has inherent risks, but that doesn't mean it's impossible (even in today's climate) to get all four apps approved.
Which risks?
Here are my priorities:
1) Sapphire AND Ink Bold.
2) United/Continental.
3) Freedom (for the 10 pts. per transaction). As I mentioned before I already own a BA card (less than a year) and Priority Club card (at least 7 years) to maybe convert into freedom (although I much rather get the bonus, as long as it doesn't take away other bonuses, obviously).
What would be my course of action?
PS. Credit score above 750.