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This model was made by extruding 2 Bezier curves. I tried joining them at the four vertices but the subsurface doesn't work.

This is the result. I've tried to delete and recreate the vertices several ways but I always get the same or similar result.

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    $\begingroup$ This looks like the normals are flipped on half of the model, so subdivision surface claims the outermost edge as the interpolated midpoint between these faces. $\endgroup$ – Allen Simpson Feb 22 at 16:11
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Face orientation (normals) was the problem. Once I rebuilt the parts from scratch and made sure to extrude both wireframe in the same direction the issue went away.

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