Multiple 74HC595 chained together won't light all LEDs
From what you said about the resistor, figuring nearly-to-the-rail output from the CMOS 74hc595 and about 1.25V diode drop, each segment probably is pulling about 7mA. If you get two full displays going that's starting to add up. Just 14 segments at this current (not even two full digits worth) you're already breaking 100mA. Not to mention that if you are powering the 74HC595's off the same 5V source that's still more current, though thankfully not too much since it is a CMOS device.

I'd have to look at the specs of the arduino to be sure, but it wouldn't be surprising if the tiny little regulated supply on the arduino isn't up to it. Even if it were rated for it, I'd probably try to at least feed the 74HC595's with a separate 5V supply. To use a separate supply, be sure to tie the grounds together. 

If you plan on running 6 digits, accounting for all segments on, you'd want something with at least 500mA or even better 1000mA. It's better to have the amp-capacity and not need it, than need it and not have it.