MinQty is softer than all-or-none. You accept partial fills, but only meaningful ones. It is common on dark and midpoint venues where small "pinging" orders are used to detect resting size.
The protection costs you fills: any counterparty smaller than your minimum is skipped entirely, so a high minimum in a thin name can mean no executions at all.
Example: you rest 40,000 shares on a midpoint-peg with MinQty 5,000. A 200-share ping from a detection algorithm is rejected, which is the point. A genuine 8,000-share institutional order trades with you. Set MinQty to 25,000 and even that counterparty passes you by.
Related: all-or-none, midpoint-peg, hidden-order