You should be able to get a few cards approved in a short period of time. I just did HHonors and Delta. The Delta one is still pending because I applied for it in less than 24 hours from the Hilton card. Other members have reported submitting a few applications at once with different browsers.

I asked AMEX about signing up for multiple cards and was told that multiple applications within 24 hours usually will need to be reviewed. I wanted to get the SPG card also and was told that shouldn't be a problem even though i just applied for two of their cards the last few days. I might give them a call later to find out what's causing the delay for my Delta Gold card.