Yes. Policies vary of course but the 3 I've activated (cr cards) all exclude any Act of war, unsurprisingly .But if say you have medical expenses unrelated to war then are you covered? Sharing my readings of pds, my bolding,
1. Macquarie (Chubb) - No. No coverage for anything in "Do not Travel" level countries. Doesn't specify "Reconsider your need to travel" level countries so presumably ok there. (P54 1. i.)
2. St George (Allianz) - unclear but arguably Yes. Exclusion is if "your claim arises because you...did not follow..."Do not Travel" or "Reconsider your need to travel" levels. So it might come down to the reason for the smart traveller warning, if you're in hospital because of war shrapnel or because of normal medical condition. (P31)
3. MyCard (Allianz) - unclear but arguably Yes. Exclusion is if "expense caused by or arising from...you not following advice..." from Smart Traveller DnT or Reconsider... (P14)
Unfortunately (?) I'm not a lawyer but would probably need one to argue the case.