So, a) this actually is more common than you think it is and b) the cause is generally not the website per-se. Many websites are front-ends on top of non-web systems. This particular system is likely a legacy system dating back to e.g. the 1970s, and once available only to employees. They often had e.g. their maintenance schedules determined by the schedules of the office employees who worked there, and may -- for ongoing operation -- require a daily "batch job" scheduled outside of office hours which can take non-deterministic amounts of time and, crucially, cannot be guaranteed safe until the program (whose architect is long-since retired) terminates. Given the potentially severe consequences of shaking up the system (which is pretty bulletproof if you use it exactly as envisioned by the architects in e.g. the 1970s), the simplest solution is to refuse to broaden the acceptable hours for active operation from the ones that are printed in the docs from the 1970s.
Rest assured, in 2050 there will be people cursing our names, too.