  We present measurements of 1/f frequency noise in both linear and
Josephson-junction-embedded superconducting aluminum resonators in the low
power, low temperature regime - typical operating conditions for
superconducting qubits. The addition of the Josephson junction does not result
in additional frequency noise, thereby placing an upper limit for fractional
critical current fluctuations of $10^{-8}$ (Hz$^{-1/2}$) at 1 Hz for
sub-micron, shadow evaporated junctions. These values imply a minimum dephasing
time for a superconducting qubit due to critical current noise of 40 -- 1400
$\mu$s depending on qubit architecture. Occasionally, at temperatures above 50
mK, we observe the activation of individual fluctuators which increase the
level of noise significantly and exhibit Lorentzian spectra.
