Bug introduced in 9.0 or earlier and persisting through 11.0.1 or later

From the Interval documentation:

For approximate machine- or arbitrary-precision numbers x, Interval[x] yields an interval reflecting the uncertainty in x.

$Version (* 10.4.0 for Microsoft Windows (64-bit) (February 26, 2016) *) Interval @ $MaxNumber === Uncompress @ Compress @ Interval @ $MaxNumber (* True *) IntervalMemberQ[Interval @ $MaxNumber, $MaxNumber] (* True *) IntervalMemberQ[Uncompress @ Compress @ Interval @ $MaxNumber, $MaxNumber] (* False *) 

What is going on here? What happens with the interval after the compress-uncompress round-trip?