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 maximum amplitude of the background, we suppose that all known supermassive black holes are the result of equal-mass mergers. This yields a maximum characteristic signal amplitude at a period of one year $A_{\rm{yr}}<1.8\times 10^{-15}$, which is comparable to the pulsar timing limits. To derive the {\em minimum} amplitude requires an electromagnetic observation of a 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. (read more)