After reading @OrganicMarble's answer and the link there as well, I looked at the TLEs for the Solar Maximum Mission in 1989 (1980-014A, 11703) and while the altitude loss in March does not seem to be in any way dramatic overall, there is in fact a drop in the average altitude at the time.
I've propagated the TLEs for one complete orbit starting at the epoch of each TLE and plotted maximum and minimum values, and for the second plot also included the time-averaged altitude (averaged over one orbit, at epoch). At about day 73 there is indeed a sudden drop in altitude, although it seems more like only a 1 kilometer "brick wall".
The loss in altitude is not the problem as much as the fact that even a tiny but unexpected change in altitude will change the period, and so the object will move faster and will not appear where it is expected. This advance in phasing is probably a better way to think about the cause of the sudden loss of tracking, rather than the altitude itself.
A drop of 1 km will shorten a 5595 second period by about 1.2 seconds. At a velocity of 7.65 km/s, a 1 kilometer drop in altitude advances the satellite by about 6.1 km per hour, or 145 kilometers per day. So it won't be anywhere near where it is expected the next time one looks for it.