Let's review what "highly unlikely" means a bit.
Looking at baseball-reference.com, the number of triples per game has historically declined, from hovering in the .4s and .5s in early history to the current rate of under .2. I'll use .35 as a rough historical average, because I don't feel like doing detailed calculations to find the historic average.
By similar reasoning, I'll declare 1.35 as the historical WHIP. It's probably not perfect, but we're just going with rough estimates here. Figuring that rare events like HBP cancel other rare-ish events like double plays, and we can then conclude 4.35 batters per half inning as reasonable for a rough calculation.
This yields 78.3 batters per game. Thus we get a triples average of about .00447. That is to say, with a typical at-bat until perfectly neutral conditions, a triple has a probability of happening 0.447% of the time. Five triples in a row happens at a rate of 1.78*10^-12. Six triples in a row happens at a rate of 7.9*10^-15. So, very unlikely...But that's for any given sequence of six batters.
I'm playing in MLB116256 right now. I'll use that as a rough estimate of the leagues that have been played. Let's further assume the standard 24-team, 162-game schedule is a perfect average for simplicity -- smaller and larger leagues cancel. So that's 12 games per timeslot, 162 timeslots for a total of 1944 games per regular season. Let's call 4, 5.5, and 5.5 average playoff series lengths, and we get an average postseason total of 32.5 games. I'll lop it to 32 to account for some shorter seasons, as larger leagues don't generally have expanded playoffs. So that's 1976 games total, on average, per league. That's 229,721,856 games played. Use the 78.3 batters per game figure, and... 17,987,221,324.8 is the number of expected at-bats. Obviously you can't have a partial at-bat but such is the nature of estimation. In scientific notation, this is about 1.8*10^11. So we're still looking at a decent-sized gap of 10^4, or about 10000 times for six triples, but only about a factor of 10 for five triples. But this is all assuming perfectly neutral conditions. If a team builds of all deadballers, plays in Petco or a similar park, and is going up against a homerun-suppressing deadball pitcher, this factor can start to shrink tremendously. I would conclude that at one point in SIM history, a five triple run might well indeed happen. Six triples is a bit harder, admittedly, but it's still possible especially if we design to hit triples.