I can only speak for seasons 4 and 5, as I don't have experience with the earlier seasons.
For season 4, no problem. There are two pretty clearly-defined story arcs, though one is more clear to the PCs than the other. There's a good roster of recurring NPCs (and their accompanying factions) and the overall quality and variety of the missions is pretty good. There's a few with some wonky bits ("What do you mean, the job starts in 15 minutes getting a guy from the building across the street?") but few spots that are really problematic. You can also fold in the "Elven Blood" mini-campaign, which sees the runners make a side trip to Tir Tangire. The one gotcha, depending on how you want to play it, is that the end of the season has something of a fixed point in time - the next-to-last is Election Day, November 2074, which wraps one of the two arcs. The finale takes place a month later.
You've got a lot of material to work with here: 13 adventures (00 to 12) in Season 4 proper, the prequel scenario "Copycat Killer" (originally a convention-only mission) which comes packaged with SRM 04-05, and the option of adding in the five Elven Blood adventures. You could also intersperse the adventures from "Firing Line" and "Sprawl Wilds", which are dual-statted for both 4th and 5th editions. Two of these (Humanitarian Aid and Burn) play very well into the Season 4 story arcs. The others make nice side trips, with the possible exception of "Lost Islands Found", which I frankly don't grok.
Season 5 is a bit more problematic. First, aside from the dual-statted adventures from "Firing Line" and "Sprawl Wilds", there are only three adventures published for Season 5. There are 16 convention missions from 2013 and 2014 as well, but these are not publicly available. And while there are some really interesting characters and situations in these three (and the next two of the first batch of six), the shape of the overall story arc is somewhat lacking. The individual adventures are good, it's just the thread that binds them isn't nearly as strong.
This doesn't mean you couldn't use these as a launching point, but does mean you don't have very much to go with unless you're an active Catalyst Demo Agent and have access to the Season 5 material that's not published yet.