I don't entirely agree with the previously given answers. It depends on the landscape structures being present in the DEM.
Equidistance in that context means vertical distance and not horizontal x or y distance. As stated before, coarse resolution can result in inaccurate contour lines, simply because the height measurements are not as dense as in a DEM with higher resolution. But in plain areas, meaning little variation concerning height values, it is valid to choose equidistance values smaller than the given DEM resolution. Imagine a landscape ranging from 0 m to a maximum height of 20 m. Maybe a beach with reoccuring peaks of dunes (mind the max height of 20 m) is a nice picture. The generation of contour lines based on a DEM with 30 m resolution would result in just one contour line - if you would choose an equidistance value of 30 m or higher. In that case you miss out morphological information in your derived contour line dataset. That doesn't mean you can create a highly detailed contour line dataset by setting the equidistance parameter to 1/10th of your DEMs resolution, although the result might look like it. But if the dune structure is large enough to be captured by the spatial resolution of the DEM, you surely can set the equidistance parameter lower than the DEM resolution in order to maintain the dune structure in your contour line dataset.
As the lower limit for the equidistance parameter I would therefore rather suggest the height accuracy of the procedure how the DEM has been created, which is about 6 m for the 30 m SRTM DEM (according to a quick check at Wikipedia). The same applies to a laserscanning derived DEM with a resolution of 1 m, where you could theoretically create contour lines with an equidistance of 0.5 m. That being said, in many cases you would probably want to choose your equidistance parameter higher than DEM resolution anyway because of height accuracy, scale or memory concerns. In the end its always good to compare the contour line output with the original DEM.
Create a 10 m contour lines dataset from a 30 m DEM, but don't expect it to be more detailed than your DEM.