Trixie doesn't so much get in trouble for boasting as she does for lying. Had she not told a fib about beating an Ursa Major, there would had never been anything to prove her wrong.
Despite the ponies' hypocrisy, the show does punish them for challenging Trixie. Only Twilight successfully upstages her, but indirectly and without much intent of humiliating her, giving off the Aesop of not going around picking fights or looking for trouble. A person's bad attitude will eventually come back to haunt them.
Anvilicious: Whatever way the Aesop is taken, it seems Ponyville does not take kindly to boasting. Or lying.
Broken Base: While Trixie herself was well-received in her majority, the episode itself is polarizing, due to the reasons listed under Designated Villain. Some love the episode specifically because it introduced Trixie and just like it for its own merits, others hate it for the mentioned problems.
Designated Villain: Trixie. While she's arrogant and boastful, one could easily chalk that up to her being a magician, an occupation for which being a Large Ham is largely demanded. She humilates several of the Mane Six, but only because they take up her challenge and try to upstage her, making themselves targets; and as Rainbow Dash herself lampshades, she and Rarity aren't always the most humble of ponies anyway. Most infamously, Trixie had little to do with the Ursa minor attack, it was Snips and Snails who brought it to Ponyville on their own, Trixie tried to fight it off but didn't have the power.
Draco in Leather Pants: This episode had fans protesting Trixie was just a simple traveling entertainer whose audience was being unnecessarily mocking and hostile, that she was justified in humiliating Applejack, Rarity, and Rainbow Dash in front of their friends and neighbors, and that she was unjustly punished at the end of the episode, largely ignoring that she's a boastful, lying egomaniac both on- and off-stage. She did seem guilty about one of her lies indirectly leading to an Ursa Minor attacking Ponyville and admirably tried to fight it off, but after the threat is dealt with by Twilight Sparkle, she's right back to being all bluff and bravado.
Ensemble Darkhorse: Trixie is extremely popular thanks to her boastfull and energetic attitude (akin a the real life performers) even if she's a arrogant Miles Gloriosus, her performance before several of the Mane 6 and ironically the Ursa Minor incident only made her more sympathetic.
Fridge Brilliance: It's fortunate for the episode's aesop that Fluttershy wasn't around when Snips and Snails brought the Ursa Minor to town, seeing as how she's great with all animals except dragons.
In this episode, Twilight's genuine magic beats out Trixie's parlor tricks. Two seasons later, Twilight's parlor tricks beat out Trixie's genuine magic. She just cannot catch a break.
Rarity's disgust at having green hair. When Season 4 introduces the "friendship rainbow" which incorporates the colors of the Mane Six's coats, Rarity's white occupies the middle position in the spectrum where green would normally go.
Unintentionally Sympathetic: Trixie really doesn't do anything wrong at the start of the episode short of exaggerating her accomplishments for the sake of her show (not exactly uncommon for performers). The show seems to expect us to root for the Mane Six who are essentially heckling her. And then the ursa minor attacks and Trixie, clearly frightened of it, tries to fight it off and is forced to admit her boasts were hollow.