It means using up the available resources, or perhaps using them to the point where they are not functioning well. In this context, maybe long waits to go to trial, or a shortage of lawyers to represent clients.
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Anonymous... and the cumulative pressures they are putting on the judiciary are straining it as never before.... and the cumulative pressures the traffickers are putting on the judiciary are making the situation more and more difficult/stressful for the judiciary as never before.