To be precise, a character with 10 base hit points could theorethically stay alive until he has lost 60 hit points - after this, he is irrevocably dead.
However, any fantasy melee fighter is going to have more hit points than that. Hit Points in 4e are now based on ST, which costs 10 point per level. And Extra Hit Points only cost 2/level now - so depending on how many you can talk your GM into allowing you, the character can keep standing for a very long time.
Let's take one of the melee fighters from my GURPS Eberron campaign. He has ST 15, 5 levels of Extra Hit Points, and HT 13.
Once he has lost 20 hit points, he will have to start rolling his HT versus unconsciousness every round to remain standing - a 83.8% chance of succeeding. Once he has reached -20 hit points, he will have to roll versus HT again to see if he dies from his wounds. The same is true at -40, -60, and -80 hit points - and at -100 hit points, he is irrevocably dead. The chance that he will make it that far if he continues to get injured is 49.3% if he keeps on making is rolls versus death.
However, the odds are that he will fall unconscious long before that, and it is rather easy to kill an unconscious foe. Still, in theory he could survive a loss of 119 hit points - not bad for a supposedly "lethal" game...