Not really, no. Generational ships might make colonizing the nearest star systems possible, but even colonizing our own galaxy would require some kind of suspended animation. The milky way is between 100,000-200,000 light years in diameter so even at the speed of light, you're looking at a travel time that is ~33-66% of the time that humanity has even existed(homo sapiens are currently estimated to have become a distinct species 200,000-300,000 years ago)... just to go to ONE star system out of the hundreds of BILLIONS that exist in our galaxy. You're gonna need generational ships so self-sustaining and capable that the generation that actually arrives at the destination will have long forgotten the point of the trip and might not want to leave the comfort of the ship.
Still, colonizing our own galaxy is at least theoretically possible, given enough time. The real filter is just how unimaginably large the universe is. The vast, VAST majority of the observable universe is FOREVER out of our reach, as it is expanding away from us faster than the speed of light. Then there's the unobservable universe, which could literally be infinitely bigger than the observable universe for all we actually know.