The order-parameter susceptibility of dilute Ising models with random fields and dilute antiferromagnets in a uniform field are studied for low temperatures and fields with use of low-concentration expansions, scaling theories, and exact solutions on the Cayley tree to elucidate the behavior near the percolation threshold at concentration pc. On the Cayley tree, as well as for d>6, both models have a zero-temperature susceptibility which diverges as ln(pc-p). For spatial dimensions 1<d<6, a scaling analysis shows that the susceptibilities of the two models exhibit the same divergence, (pc-p)-(p-p)/2, where p and p are percolation exponents associated with the susceptibility and order parameter. At d=6, the susceptibilities diverge as ln(pc-p)9/7. For d=1, exact results show that the two models have different critical exponents at the percolation threshold. The (finite-length) series at d=2 seems to exhibit different critical exponents for the two models. At p=pc, the susceptibilities diverge in the limit of zero field h as h-(p-p)/(p+p) for d<6, as lnh9/7 for d=6, and as lnh for d>6.