  To establish a unified framework for studying both discrete and continuous
coupling distributions, we introduce the {\it binomial} spin glass, a class of
models where the couplings are sums of $m$ identically distributed Bernoulli
random variables. In the continuum limit $m \to \infty$, the class reduces to
one with Gaussian couplings, while $m=1$ corresponds to the $\pm J$ spin glass.
We demonstrate that for short-range Ising models on $d$-dimensional hypercubic
lattices the ground-state entropy density for $N$ spins is bounded from above
by $(\sqrt{d/2m} + 1/N)\ln2$, and further show that the actual entropies follow
the scaling behavior implied by this bound. We thus uncover a fundamental
non-commutativity of the thermodynamic and continuous coupling limits that
leads to the presence or absence of degeneracies depending on the precise way
the limits are taken. Exact calculations of defect energies reveal a crossover
length scale $L^\ast(m) \sim L^\kappa$ below which the binomial spin glass is
indistinguishable from the Gaussian system. Since $\kappa = -1/(2\theta)$,
where $\theta$ is the spin-stiffness exponent, discrete couplings become
irrelevant at large scales for systems with a finite-temperature spin-glass
phase.
