Why time decay and convexity usually oppose

Why do an option's theta and gamma usually have opposite signs? Is this always the case?

From the Black-Scholes PDE, $\Theta = rV - rS\Delta - \tfrac12\sigma^2S^2\Gamma$, so for a hedged book convexity has to be paid for with decay, but not always. When the $r(V - S\Delta)$ term dominates, both can be positive: a deep in-the-money European put has $\Gamma > 0$ yet positive theta, and similar exceptions arise with large dividends or for some digitals and barriers.