We study the cosmic variance limit on constraining
primordial non-Gaussianity for a variety of theory-motivated
shapes. We consider general arguments for 2D and 3D surveys, with a
particular emphasis on the CMB. A scale-invariant N-point
correlator can be measured with a signal-to-noise that naively
scales with the square root of the number of observed modes. This
intuition generally fails for two reasons. First, the
signal-to-noise scaling is reduced due to the blurring of the last
scattering surface at short distances. This blurring is caused by
the combination of projection and damping, but the loss of signal is
not due to exponential decay, as both signal and noise are equally
damped. Second, the behavior of the N-point correlator in the
squeezed and collapsed (for N > 3) limits can enhance the scaling of
the signal-to-noise with the resolution, even with a reduced range
of momenta probing these limits. We provide analytic estimates for
all N-point correlators. We show that blurring affects
equilateral-like shapes much more than squeezed ones. We discuss
under what conditions the optimistic scalings in the collapsed limit
can be exploited. Lastly, we confirm our analytical estimates with
numerical calculations of the signal-to-noise for local, orthogonal
and equilateral bispectra, and local trispectra. We also show that
adding polarization to intensity data enhances the scaling for
equilateral-like spectra.