It depends on how your RLC bridge measures inductance. One measurement at a single frequency isn't sufficient. An advanced test instrument may take multiple readings at different frequencies. Since it's not possible to separate the self-capacitance from the inductance during the measurement, most simple inductance meters will give you an inductance value that's slightly lower than the "true" (DC) inductance value because it can't compensate for self-capacitance. By measuring at two frequencies, you have enough data to determine both the self-capacitance and the "true" inductance.
To elaborate on the measurement technique I described earlier:
Suppose you connect various values of capacitance across the terminals of the inductor, and then measure the resulting resonant frequencies. The reciprocal of the square of the frequency will be directly proportional to the total capacitance. So, if you plot 1/F^2 vs. C on a graph, you will get a straight line. If the coil had no self-capacitance, then the line on the graph would pass through the origin, [0,0]. But because of self-capacitance the line will actually cross the C axis at –Cs, where Cs is the self-capacitance.
(The formula that was given by MrAl is a simplification of the general formula that results if you pick the external capacitance values such that one is twice the value of the other.)
Referring back to the straight line 1/F^2 vs. C relationship, the true inductance is proportional to the slope of the line, and is given by:
L=1/4π^2*(1/F1^2-1/F2^2)/(C1-C2)
So, two measurements are required in order to determine both the self-capacitance and the true inductance. However, it's better still to do more than two measurements using multiple external capacitance values and then do a linear regression. This gives much better accuracy, and in addition, it allows you to see just how straight the the line is, and consequently confirm that the theory is valid. I have a spreadsheet on my personal website that uses the linear regression method, if you're interested:
http://electronbunker.ca/Extras/Downloads/LoopSelfCap.ods