First, what is hangover? Hangover is a combination of symptoms, such as nausea and headache, which occur several hours after stopping drinking - that is when the blood alcohol levels start to approach zero; more symptoms and proposed mechanisms are listed here.
Question: Is there any correlation between the quality of the consumed alcoholic beverages and the likeliness of getting a hangover?
Methanol in alcoholic beverages may cause hangover. Alcohol Hangover: Mechanism and Mediators "Possible contributing factors to hangover: compounds other than alcohol in beverages, especially methanol." NOTE: This is their opinion, not a study result.
Alcoholic beverages high in congeners--byproducts of alcohol fermentation--(whisky, rum, red wine)--seem to cause hangover more often than those with less congeners:
Whisky vs. vodka: a review In this 2010 review of studies, authors agree that "...highest congener beverage (bourbon) results in more severe hangover ratings than does the beverage with essentially no congeners (vodka), although ethanol effects per se had a considerably stronger effect on hangover than did congener content."
Alcohol type and hangover severity Reported hangover severity (in decreasing order) after drinking of various beverages: brandy, red wine, rum, whisky, white wine, gin, vodka, beer, ethanol diluted in orange juice. According to the article, hangover severity is related to amount of congeners.
My conclusion: Hangover may be caused by ethanol and by congeners, but, from studies available, there is insufficient evidence that alcohol beverage quality ("well made", price) would be related to hangover.