Something like

diff --git a/lib/matplotlib/tight_layout.py b/lib/matplotlib/tight_layout.py
index 1f61d95e9..85f8a26d0 100644
--- a/lib/matplotlib/tight_layout.py
+++ b/lib/matplotlib/tight_layout.py
@@ -199,7 +199,8 @@ def auto_adjust_subplotpars(fig, renderer,
     if rows > 1:
         vspace = (max(sum(s) for s in vspaces[cols:-cols])
                   + vpad_inches / fig_height_inch)
-        v_axes = (1 - margin_top - margin_bottom - vspace * (rows - 1)) / rows
+        v_axes = max((1 - margin_top - margin_bottom - vspace * (rows - 1)) / rows,
+                     0.01)
         kwargs["hspace"] = vspace / v_axes

     return kwargs

(and likewise for vspace) "fixes" the issue but the subplots are still tiny and squished (fundamentally there's just not enough space) so as @NelleV suggested an error may be better.

(it's max(..., 0.01) and not max(..., 0) because matplotlib unsurprisingly doesn't like zero-sized axes...)

I've stumbled onto this when plotting a 3x7 gridded figure in A4 paper. Has this been pushed?
(Matplotlib 2.2.2)

@jklymak this looks like #4413 on the opposite axis.

@amyspark This happens if the title or the axes labels are too long. There is (and probably will be) no fix for this - mainly because there is no canonical action to take here; if the elements in the figure are too large, they are too large. (However in future versions, a warning will be emitted in such cases, see #10915)

You need to decide for yourself what outcome you wish to have. Either make sure the labels or title are short enough not to overlap the axes by using a smaller fontsize or wrapping the text. Or, if you need those long labels, call tight layout before setting the long text to labels.

Closed by #10915.