As Catija mentioned in her answer, Rapunzel shows Flynn a lock of brown hair in the film which never increased after Mother Gothel cut them:
If that single hair streak never changed in her whole life then we can safely assume that her hair will never grow back naturally. But she didn't really lost the power as she was able to revive Flynn Rider by her magical tears.
But eventually her magical hairs did came back in the sequel Tangled: Before Ever After:
From Wikipedia synopses: (which contain spoiler from the film)
Accompanied by Maximus and Fidelio, Rapunzel and Cassandra sneak out of the castle and venture into the woods. Climbing over the wall and crossing an old bridge, Cassandra takes the princess to the former location of the healing flower that saved Rapunzel and her mother when the latter fell ill during her pregnancy. Mysterious thorn-like rocks have appeared in the area, and not only are they ominous, but also unbreakable. As Rapunzel examines the area further, she is struck by magic the moment she lays a finger on one of the rocks. Part of her hair begins to glow, and more of those mysterious rocks suddenly sprout from the ground. Rapunzel and Cassandra run back to the old bridge, chased by a trail of sprouting rocks. Running far ahead, Rapunzel's hair begins to glow even brighter, and as she makes it out of the woods her long blonde hair that was cut grows back. No time to figure out what had just happened, Rapunzel and Cassandra cross the bridge, only for Rapunzel's hair to get stuck. The bridge beneath them begins to crumble apart, but with Maximus' help, the girls manage to free Rapunzel's hair and make it safely to solid ground before the bridge finally collapsed. As morning rises on the day of her coronation, Rapunzel and Cassandra sneak back into the castle and waste no time trying to get rid of the hair. However, they discover that Rapunzel's newly restored hair is unbreakable like the thorns.
Remember this TV movie also serve as the pilot for sequel TV show Tangled: The Series:
Since being freed from Gothel and reunited with her birth parents, King Frederic and Queen Arianna of Corona, Rapunzel adjusts to her life as their daughter and continues to have different misadventures with her boyfriend Eugene Fitzherbert, Pascal the Chameleon, Maximus the Horse, and her new handmaiden Cassandra, to uncover the mystery of her new 70 feet-long blonde hair's return.