Never power an LED off 3.3V all by itself. Doing so runs the risk of burning out the LED (due to excessive current draw... diodes have exponential I-V curves, so small changes in voltage = big changes in current) and, because of the excessive current draw, browning out Tessel's 3.3V rail, which will undowbtedly play havok with the CC3000.
The right answer is to put a current limiting resistor in series with the LED to limit the current the LED draws to, say, 5mA for a "standard" 5mm diode (less for a smaller surface mount part). Assume ~1.6V of drop across the LED; it's better (safer) to assume a smaller drop than a larger one.
On a similar note, I don't know how you're controlling the LED, but it's a common misconception that GPIO pins can source meaningful current. Don't bank on more than ~5mA from any GPIO pin. If you short the a GPIO pin to ground or use it to drive a heavy load, you run the risk of burning out the GPIO pin in question in addition to "upstream" consequences with the power system at large.
...Anyway, apocalyptic warnings aside, 24mA is no big deal for Tessel's power plant, but as stated above, it's far too much for a poor little GPIO. Grab a resistor in the 300-600 Ohm range if you're driving an LED with GPIO directly.
The only other thing which comes to mind as a possibility, however unlikely, is how and when you turn the LED on and off. You may have better luck if you separate the state changes from radio transmission in time by as little as a millisecond.