  Using optical potentials fitted to a comprehensive set of strong interaction
level shifts and widths in $K^-$ atoms, we predict that the $K^-$ atomic levels
which are inaccessible in the atomic cascade process are generally narrow,
spanning a range of widths about 50 - 1500 keV over the entire periodic table.
The mechanism for this narrowing is different from the mechanism for narrowing
of pionic atom levels. Examples of such `deeply bound' $K^-$ atomic states are
given, showing that in many cases these states should be reasonably well
resolved. Several reactions which could be used to form these `deeply bound'
states are mentioned. Narrow deeply bound states are expected also in $\bar{p}$
atoms.
