Yes. We have ample evidence from both education writing in the period, and from fiction about schooling, to believe that bullying was much the same in the 19th century as it is today. The epithets have changed but, social media notwithstanding, the method hasn't.

In Some Thoughts Concerning Education (1693) John Locke comments darkly on the social pressure that exists between boys at school, calling it a "hazard" to "innocence and virtue". Many of the education thinkers who came after him were critical of schooling on any scale large enough to prevent every child having a personal relationship with the teacher, in part from the fear that larger schools created a lawless space (when the children were not supervised) where they would bully each other, fight, steal etc. Ironically, the author, philosopher, and children's publisher William Godwin argued (in 1797's The Enquirer) that this space, away from the teacher's eye, was necessary to teach children how to socialise with peers.

Godwin depicts school (and college) bullying in two of his novels: Fleetwood (1805) and Mandeville (1817), although both novels are set in earlier times. In both cases, boys conspire to humiliate the individual who has not fallen in step with everyone else. Fleetwood's sadistic "puppet trial" (it needs to be read to be believed) leads to the outsider's suicide. In Mandeville it is the protagonist who believes he is being bullied, contributing to a lifelong paranoia.

The most famous literary bully of the 19th century though, is Flashman - the villain of Thomas Hughes' Tom Brown's School Days (1857). We have strong reasons to think that Hughes' novel is a fictionalised memoir of his (and his brother's) time at Rugby School in the 1830s. Flashman is referred to as a bully from his first appearance, and the truism about bullying being cowardly is repeated here too. Flashman is an unsophisticated villain though, he mostly wants younger boys to act as his servants (a privilege only usually extended to the final year pupils) and uses physical violence to coerce them - he holds Tom up to the fire to be burned because he refuses him a ticket in a sweepstake. Hughes emphasises Tom's resilience and refusal to cave in to violence - Flashman is eventually expelled when he is caught drinking. The preface to Hughes' sixth edition included a letter from a school contemporary that refers to the "grief and misery" of bullying that destroys health and character for life.

How people viewed bullying changed gradually from the 18th century to the 19th, but there was a persistent debate between advocates of "toughening up" and those who regarded bullying as a moral danger. If you have access, you can read a survey of it in Katharine Kitteridge's paper here.

Most references to bullying seem to be discussions of physical terror, as in the case of Flashman, but both of Godwin's examples are elaborate psychological campaigns.