A near miss is a polyhedron that's almost a Johnson solid. Without formalizing what that means, we might ask that one should be able to construct the polyhedron out of paper (or some construction toy), using only regular faces, without noticing the discrepancy.
To our knowledge, there has never been a rigorous study of near misses, though one would imagine they were looked at carefully in the classification of Johnson solids. If anybody knows of such a study, please let me know.
Click on the image for a larger version. Sorry about the dearth of WRL files.
O(*,3,*,[2]): 8 enneagons, 6 squares, 24 almost equilateral triangles.
The notation given is from our Symmetrohedra paper. 

I(1,2,*,[2]): 12 pentagons, 20 hexagons, 60 almost equilateral triangles.  
I(2,*,3,e): 12 decagons, 30 hexagons, 20 equilateral triangles, 60 almost squares.  
I(2,4,*,e): 12 decagons, 20 dodecagons, 60 almost
equilateral triangles (these are actually trapezoids with very
narrow tops), 30 very narrow rectangles.
This solid is very similar to a truncated buckyball. One imagines that by fiddling with it a bit, one could sacrifice the perfect regularity of the decagons and dodecagons in exchange for eliminating the narrow rectangular faces, resulting in a near miss. 

[no notation]
A solid constructed by inscribing a regular 11gon in every face of a pentagonal icositetrahedron. The result is a little loose, but there's room for improvement by starting with a differentlyparameterized base solid. Alex Doskey was kind enough to contribute WRL files for this solid: Solid, Leonardostyle. 