will always periodogram produce correct frequencies?
A periodogram created from a finite number of samples using an DFT is not really a "correct" frequency estimator, as much as it is a (leaky) frequency energy binning sieve, where the width of each bin is set by the DFT length and the window function applied.  

The width and spacing of each bin may or may not be narrow enough for you to consider an error of up to half that size to still result in a useful frequency estimate.  Various interpolation methods that post-process DFT results may allow more precise estimates of spectral peaks than the bin spacing.