In 2-dim regression y = Xb the question is often whether or not b = (b1,b2) is significantly nonzero. This is usually decided by checking whether the corresponding confidence ellipse, E, for b contains the point (0,0).
Associated with E are the two marginals for b1 and b2, and corresponding single confidence intervals I1 and I2.
Consider a case where (0,0) is outside of E but 0 is inside both I1 and I2, e.g. when E is rather flat and slightly off (0,0). With respect to the original regression question: Is (b1,b2) significantly nonzero but neither one of b1 and b2 is?