For decades, forensic statisticians have debated whether searching large DNA databases undermines the evidential value of a match. Modern surveillance faces an exponentially harder problem: screening populations across thousands of attributes using threshold rules rather than exact matching. Intuition suggests that requiring many coincidental matches should make false alerts astronomically unlikely. This intuition fails. Consider a system that monitors 1,000 attributes, each with a 0.5 percent innocent match rate. Matching 15 pre-specified attributes has probability \(10^{-35}\), one in 30 decillion, effectively impossible. But operational systems require no such specificity. They might flag anyone who matches \emph{any} 15 of the 1,000. In a city of one million innocent people, this produces about 226 false alerts. A seemingly impossible event becomes all but guaranteed. This is not an implementation flaw but a mathematical consequence of high-dimensional screening. We identify fundamental probabilistic limits on screening reliability. Systems undergo sharp transitions from reliable to unreliable with small increases in data scale, a fragility worsened by data growth and correlations. As data accumulate and correlation collapses effective dimensionality, systems enter regimes where alerts lose evidential value even when individual coincidences remain vanishingly rare. This framework reframes the DNA database controversy as a shift between operational regimes. Unequal surveillance exposures magnify failure, making ``structural bias'' mathematically inevitable. These limits are structural: beyond a critical scale, failure cannot be prevented through threshold adjustment or algorithmic refinement.