Understanding FFT bin resolution
The energy bin "width" of an FFT is the width of the transform of the window used (a rectangular window by default).  

The transform of a rectangle the width of the FFT is a Sinc function (sin(x)/x) with a main lobe width equal to twice the DFT/FFT bin spacing, plus smaller ripples extending the full width of the FFT result (and actually circling around forever making it technically a Dirichlet function).  Note that the frequency response of each bin filter overlaps with others except at exact bin centers (representing exactly periodic-in-aperture pure sinusoids).

For the default rectangular window, only about 87% of a bin's passband is within 0.5 of a bin width from the bin center frequency, which is why ignoring so-called leakage might not be a good idea.  A longer FFT has a narrower main lobe bin response, with the ripples ("leakage") dying out faster.  But 2 half width Sinc functions do not sum to a single Sinc, so averaging a longer FFT will not produce identical results (except for sinusoids that are exactly periodic in both FFT lengths.)

A window other than rectangular (von Hann, etc.) will have a main lobe wider that 2 bin spacings, but, in return, far smaller ripples away from the main lobe.

The DFT/FFT resolution depends on whether you are trying to resolve closely spaced peaks as clearly separate peaks (requiring around double or more the bin spacing), or estimating the frequency of a single isolated peak with a relatively high local S/N ratio (sometimes in low noise allowing a fraction of a bin spacing in resolution after high quality interpolation).