Power issues with RFDuino
I am the designer of the RGB Shades and co-own macetech LLC. The RFDuino specifications claim only 18mA on transmit and receive, so there should not be a current draw issue from that module.

Known power issues with the RGB Shades include high current draw on startup. They can pulse about 1.3 amps (the blue flash) before your software sets them to a more reasonable level. Most USB power packs made today can handle 1.0 amps, some can supply more, but it's still common to find packs limited to 500mA...that would be one thing to check.

Since you are using your own code, please also note that our default code on the controller-included RGB Shades sets the maximum brightness to about one-quarter the actual maximum brightness of the LEDs. This is done to conserve power as well as reduce brightness to non-blinding levels. If your code is attempting to operate at normal brightness, please be aware the RGB Shades could draw up to 4 amps.

The best way to detect the problem would be to get a multimeter and compare the supply voltage when connected to your computer and when connected to the USB power pack.