There was discussion earlier in the thread about whether it's better to call recon immediately or wait and see. The following experience might suggest the wait and see approach is better:
I did a Chase 3bm for my wife (CO UA SP) with no instant approvals. When calling the automated status line, it said 2 were "7-10" (indicating chance at approval) and 1 was "2 weeks" (indicating denial). When I called, they started asking why so many cards, and then looked into wife's credit and would not approve more than one. Another call to recon was to no avail. The final letter I got said "too many requests for credit."
No clear proof, but I would try waiting it out next time.