We present X-ray and multiwavelength analyses of 32 weak emission-line quasars ( WLQs ) selected in a consistent and unbiased manner . New Chandra 3.1–4.8 ks observations were obtained for 14 of these WLQs with C iv rest-frame equivalent widths ( REWs ) of 5–15 Å , and these serve as an X-ray observational ‘ ‘ bridge ’ ’ between previously studied WLQs with C iv REW \lesssim~ { } 5 Å and more-typical quasars with C iv REW \approx 15–100 Å . We have identified and quantified a strong dependence of the fraction of X-ray weak quasars upon C iv REW ; this fraction declines by a factor of \approx 13 ( from \approx 44 % to \approx 3 % ) for C iv REW ranging from 4–50 Å , and the rate of decline appears particularly strong in the 10–20 Å range . The dependence broadly supports the proposed ‘ ‘ shielding ’ ’ model for WLQs , in which a geometrically and optically thick inner accretion disk , expected for a quasar accreting at a high Eddington ratio , both prevents ionizing EUV/X-ray photons from reaching the high-ionization broad emission-line region and also sometimes blocks the line-of-sight to the central X-ray emitting region . This model is also supported by the hard average spectral shape of X-ray weak WLQs ( with a power-law effective photon index of \Gamma _ { eff } = 1.19 ^ { +0.56 } _ { -0.45 } ) . Additionally , we have examined UV continuum/emission-line properties that might trace X-ray weakness among WLQs , confirming that red UV continuum color is the most-effective tracer .