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I'm trying to put this solid

Surface1

S1 = RegionPlot3D[z - 50 <= -2 x^2 - 2 y^2, {x, -5, 5}, {y, -5, 5}, {z, 0, 50}]

on top of this one

Surface2

S2 = 
  RegionPlot3D[x^2 + y^2 <= 50 , {x, -10, 10}, {y, -10, 10}, {z, -50, 0}, 
    PlotStyle -> RGBColor[0.15, 0.17, 0], Mesh -> None]

and it's just kind of smashing it into this

Show[
  RegionPlot3D[
    z <= -2 x^2 - 2 y^2 && z <= 8 x + y - 20, 
    {x, -10, 10}, {y, -10, 10}, {z, -100, 0}], 
  RegionPlot3D[x^2 + y^2 <= 50 , {x, -10, 10}, {y, -10, 10}, {z, -500, -100}, 
    PlotStyle -> RGBColor[0.15, 0.17, 0], Mesh -> None]]

Smashed Surface

How to combine these 3D plots correctly?

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Try this

With[{color = RGBColor[0.95, 0.93, 0]},
  Show[
    RegionPlot3D[
      z <= -2 x^2 - 2 y^2 && z <= 8 x + y - 20, 
      {x, -10, 10}, {y, -10, 10}, {z, -100, 0}, 
      PlotStyle -> color, Mesh -> None],
    RegionPlot3D[
      x^2 + y^2 <= 50, 
      {x, -10, 10}, {y, -10, 10}, {z, -500, -100}, 
      PlotStyle -> color, Mesh -> None], 
    PlotRange -> All]]

solid

Your problem was mainly not giving the option PlotRange -> All to Show. Without that, you were only getting the range for first RegionPlot3D.

I changed the coloring only to get a better looking plot for this post. It not a part of your problem. You can choose any color you like.

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Another way can be defining a single function

f[x_, y_, z_] := If[z > 0, z - 50 <= -2 x^2 - 2 y^2, x^2 + y^2 <= 50]
RegionPlot3D[f[x, y, z], {x, -10, 10}, {y, -10, 10}, {z, -50, 50}]

enter image description here

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