Over the course of his up and down life, the most important lesson Pip learns is that wealth and status are not the most important things in life. Much of his life was spent in pursuit of his so-called "great expectations" but when, for a time, he gets close to being a gentlemen, he realizes that love and family are much more important. The society people he encounters (Miss Havisham, Estella, Bentley Drummle, and the high society "wannabe" Uncle Pumblechook) are poor excuses for human beings. It takes Pip quite awhile to learn this. Meanwhile, Biddy and Joe remain true to him throughout his life, and when Pip awakes from his illness, it is Joe by his side, who has nursed him for so long, not any of his so-called "high society" acquaintances. What is more, Pip is deluded for quite awhile that the wealthy Miss Havisham is his benefactor, but when he learns the truth - that his benefactor is from the lowest rung on the social ladder - an uneducated convict - it turns Pip's world upside-down. In this way, Dickens skillfully illustrates his theme - that there is more to life than wealth and society. True wealth comes from love and family -- these are more important "great expectations".
While Pip never fulfills the original ideal that he possessed of a gentleman, I would argue that he does eventually become a gentleman, in the figurative sense of the word. When Magwitch initially (secretly) expresses his great expectations for Pip, he desires for Pip to be wealthy and to exhibit high class. For Pip, wealth and class are also what he envisions when he strives to be a gentleman. However, as Pip develops into a snob, gets himself into financial difficulties, and has to figure out a way to keep Magwitch safe, Dickens begins illustrating through characters such as Joe and Wemmick what a true gentleman is. While neither Joe nor Wemmick is incredibly rich, they both demonstrate loyalty and financial common sense, attributes which very few of the literal gentlemen (Compeyson, Jaggers, and Drummle) in the novel possess.
By the end of the novel, after Pip has made peace with Magwitch and Joe, he obtains a steady job which requires detailed skills that he would not have learned had he remained a blacksmith. Similarly, he has been able to travel internationally; he is self-sufficient, and he possesses the loyalty and class that true gentlemen of his time were supposed to demonstrate.
In regards to reasons why Pip does notexhibit the traditional characteristics of a gentleman from Dickens' time, most of those reasons are societal. For example, even though some of the limitations upon the British who were not aristocrats had started to disappear, during Pip's time, there was still a natural divide between the aristocracy and the "working" gentleman--even if that gentleman had earned more money than an aristocrat inherited. Because Pip is not born into wealth and class, societal standards are already against him when he begins his "training" in London. Pip, of course, does contribute to his own downfall by his excessive spending, but his warped view of what a gentleman is has also come from society.