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The documentation for RegionPlot3D states

You should realize that since it uses only a finite number of sample points, it is possible for RegionPlot3D to miss regions in which pred is True. To check your results, you should try increasing the settings for PlotPoints and MaxRecursion.

This is what happened in your second example, a result of the region being so thin.

As commented by george2079, you can increase the sampling in the x direction only with the option PlotPoints -> {100, Automatic, Automatic}:

RegionPlot3D[z < 1 - V^2 && ((-0.001 - 0.1) < x < (0.001 - 0.1)), 
  {x, -0.9, 0.9}, {V, 0, 1.2}, {z, 0, 1},
  PlotPoints -> {100, Automatic, Automatic}]

You might also be interested in the contourRegionPlot3D function here:

contourRegionPlot3D[z < 1 - V^2 && ((-0.001 - 0.1) < x < (0.001 - 0.1)), 
  {x, -0.9, 0.9}, {V, 0, 1.2}, {z, 0, 1}]