It depends. I've a friend who is a Greek-American, and I've been able to trace her family only as far back as her great grandparents. I have a cousin, whose father is a Norwegian-American, and using conventional web sites, such as Ancestry.com, I've only been able to trace his dad's side of the family back to his grandparents. That's not to say these records don't exist; they just haven't made it to any of the online web sites I usually access (quite possibly because they aren't in English, French, or German).
Most people who live in the United States, Canada, and the United Kingdom and have done so for many generations, however, can fairly easily trace their ancestry back to the mid 19th-century because census records go back this far. In the US, all family members (if they are white) are shown back to 1850, but most African Americans unfortunately probably aren't listed until 1870. Census records before 1850 list the male head of the family only; the first census was in 1790, and it mentions only male heads of the household, so the researcher has to have a few clues where he or she is going. I suppose the most interesting trend I've noted is that for the most part family groups moved together from other countries and within the United States. This made my ancestry easier to trace.
In some of my family's lines, I've been easily able to trace my ancestry back to the mid-17th to the early 18th-century in both England and Scotland because the parish records seem to be intact. Some of my paternal grandfather's ancestors, I've even been able to trace back to the Middle Ages because of a tie in to the House of Stuart, which came as a surprise.
Actually, I think I've lucked out because I've run across distant cousins who were members of the Latter-Day Saints Church, so they had already done of a lot of my "leg work" for me, and in many cases this includes having the actual documentation. Accordingly, participating in family web sites has been a big help, although I still have a way to go in some instances to find immigrant ancestors or to determine at what date they came to the United States.
I've also had the fortune to trace some fairly unusual last names, for example, Macquarrie, Box, and Keesee (an American corruption of the French La Cage), even if my surname is Smith. Usually, all of the above living in the United States in each case trace back to an original immigrant or at most perhaps a pair of brothers. The Smiths I've been able to trace back to about 1800 in Massachusetts, although census records note that my great great grandfather's dad was from New Hampshire. I'll probably have to be satisfied that I've traced them that far.