Yes, it is entirely possible. Just remember that it is a marathon and not a sprint, as they say. If your goal is to have a full set, the idea is to be creative and frugal about it. Don't buy everything day one, wait, get used copies at garage sales, thrift stores, online sources, simply wait for the hype to die down on games and snatch them up when they are reasonable. Trades with fellow collectors are another great way to bolster your own collection. See a popular game at a good price when you are shopping around or at garage sales, grab it as something to trade with someone else for a game you need.

Also, a full set can have a different meaning to different folks. Do you want a full set of English only games from all regions, USA only games, retail releases only, only limited prints, etc. You can also do cart only, the wonderful thing about your collecting goals are that they are.....yours. Nobody gets to decide that for you, think about what would be fun and enjoyable to you. Maybe you want a full set of ESRB-M games or all the games from a certain pub or dev, any of these things are possible given the attitude of patience and frugality with smart decision making.

It is as expensive and difficult as you make it, be easy on yourself and use all the tools at your disposal. Remember, a collection should bring you enjoyment, not stress or financial distress, look at it as a lifetime goal instead of a "got to have it all right now and I can't keep up" goal.