I've had bouts with hard starting with a '02 vintage MS390, which I think is in the same family 250,290 310, might not be but, there are some things I can think of that may or may not apply. Is this a cold start?
That might be a fuel line, not sure about the impulse line, but mine had a bad fuel line for awhile, runs/stalls, hard starting and any combination of similar problems.
When hot, I have on occasion, flooded it, thinking it cooled enough to have to choke or half choke, 2nd position on lever, and believe in those instances it was flooded, let it sit for a few minutes it will fire up. I realize its better to see if it will start no choke, then use the choke if no dice, after its been run, depends on the temperature, sometimes it cools right off and its a cold start, they are fussy that way or so mine is.
If you have to pull that many times on one of these, from my experience there is a problem.
Best case, no decompression on mine, with a clean air filter, no issues with spark plug,fuel lines or what have you, from cold, full choke, one pull, it should fire, maybe 2 if its sat awhile. from that first pull, I have been able to move the lever to the 2nd position, kept it running, then squeezed the trigger and worked the throttle, most times it cuts out, move the lever, pull and it does the same, and will keep running. Seems as that air filter,(and it does have an intellicarb) fills with particulates, its does take more to start. This one has been temper-mental with starting, and still cuts out sometimes on the descending side of the throttle, meaning, after a cut or some useage, even revving it a bit, it winds down it will cut out, like if you hit a switch, obviously fuel related, after a new fuel line was installed, sometimes you can move it, work the trigger, it will run, might just be the pick up in the tank, the position, you can make a wire hook to pull that and check the screen, I keep on in my saw tool box. One thing I am not impressed with is the barbed plug connecter, that is of some kind of spring steel and penetrates the plug wire to the coil or ignition module, that I always suspected the connection on this type, could be a problem.
I'll know soon as mine is going to be completely disassembled, have all new OEM plastic for it, and will be putting that on, replace fuel line impulse line, see if there is another ignition wire/plug connecter for it. I like the saw for the most part, plenty of power, I try to take care of it mechanically, but aesthetically from use and not having a case it shows, hopefully these problems are sorted out. I do recall it just seems to start and run better with good service and maintenance, performs great, but I won't lie, it has given me some trouble, like hard starting at times.