So this got me curious if this has ever happened before.
In around 1920 and 21 it's likely that 3 of the tallest were under construction at once with 714 Main, W.T. Waggoner, and Hotel Texas. Neil P. was also around this time but was shorter than 1914's Burk Burnett. (Though this is only if you don't consider the courthouse as a "skyscraper", as Burk Burnett is actually shorter)
Around 1929 the 2nd, 3rd, 5th, and 7th tallest buildings at the time were likely under construction at about the same time with the Blackstone, Commerce, Electric, and SInclair buildings all being completed in 29 or 30. T&P Terminal was completed in 1931 and may have been part of that as well. The Aviation building was also built at this time. The earlier mentioned 714 Was the tallest at this time.
When the Landmark building was completed in 1957, 500 W 7th was 4 years behind, I doubt there was any overlap in construction but we did see a 1st and 3rd tallest built within a 4 year time span.
The Tower was completed in 1974, and at the time the 9th and 10th Tandy Center towers were finished in 76 and 78, so there was possibly some overlap there.
It seems if you are a fan of Fort Worth skyscrapers, the time to be alive was 1921, 1929, and 1983.
Also just for fun, here's how long each building has been the tallest in order of how many years it was/is the tallest.
Burnett Plaza-33 and counting
Wells Fargo Tower-1
It looks like Burnett will soon be the king.