Yes and no. The Smithsonian Star Catalog shows that there are 3277 stars with visual magnitudes from +5.0 to +5.99m; 10081 stars with magnitudes from +6.0 to +6.99m, and 24819 stars between +7.0 and +7.99m. About half of these would be visible in the sky at any one time. Now it is common to estimate the limiting magnitude of the eye as +6.0m, but in fact it is somewhat better than this if you are young, and somewhat worse than this if you are old, with otherwise 20/20 vision. Some people can see some of the fainter stars in the Pleiades cluster, and have a limiting magnitude closer to +6.5, in which case there are perhaps 6000-7000 stars that would be visible to them. Also, our retinae have their highest sensitivity away from the fovea and by using 'averted' or 'peripheral' vision, we can do better than the canonical +6.0 limit. So, it would not surprise me if some people could see more than 6000 stars in the sky ( 3000 stars per hemisphere), especially young people who eat a lot of carrots and have spent over an hour dark adapting their eyes, far away from city lights!