  The gravitational-wave background from supermassive binary black holes
(SMBBHs) has yet to be detected. This has led to speculations as to whether
current pulsar timing array limits are in tension with theoretical predictions.
In this paper, we use electromagnetic observations to constrain the SMBBH
background from above and below. To derive the {\em maximum} amplitude of the
background, we suppose that equal-mass SMBBH mergers fully account for the
local black hole number density. This yields a maximum characteristic signal
amplitude at a period of one year $A_{\rm{yr}}<2.4\times 10^{-15}$, which is
comparable to the pulsar timing limits. To derive the {\em minimum} amplitude
requires an electromagnetic observation of an SMBBH. While a number of
candidates have been put forward, there are no universally-accepted
electromagnetic detections in the nanohertz band. We show the candidate 3C 66B
implies a lower limit, which is inconsistent with limits from pulsar timing,
casting doubt on its binary nature. Alternatively, if the parameters of OJ 287
are known accurately, then $A_{\rm{yr}}> 6.1\times 10^{-17}$ at 95\% confidence
level. If one of the current candidates can be established as a bona fide
SMBBH, it will immediately imply an astrophysically interesting lower limit.
