Yes. I have not played the Multiplayer this entire playthrough and I achieved a Total rating of 12201. After doing the Armax Arsenal Arena countless times, the rating dropped to 7926. After checking it again this morning to copy over the War Assets, it skyrocketed back over to 12201, and those assets I previously typed are the amounts it showed.
I have not played the Multiplayer this entire playthrough
That kinda implies that you did play Multi, just not during that particular playthrough. Any promotions attained in the past are automatically added to your War Assets for any Shepard, even new ones.
Yet, I have not. The last time I played the entire Mass Effect Franchise, including 3, was back in May of this year. I had played the multiplayer extensively during this time, as well. Afterwards, I had to stop as my priorities were focused in other areas. I started replaying the franchise again in mid-September.
Well, you say that you have played Multi extensively during May. Any promotions you attained then are stored server-side, so any new Shepard created after that period (even if the game was uninstalled) profit from it.
I had a situation months back with a really nasty virus, which led me to delete my old account on my pc (the virus was account-bound). As a result, I lost my saves for the Mass Effect trilogy (and my achievements for 2 and 3, strangely). I recently installed ME3 again and my new Shepard would still profit from old promotions.
Also, you gain 75 War Assets per promotion, so it can increase fairly fast since leveling in Multi is really easy.
You might want to check how many promotions you have gained ;)
'Screw the rules, I've got money!' - Seto Kaiba, YGO Abridged 'Requiescat in Pace' - Ezio Auditore da Firenze, Assassin's Creed 2/Brotherhood