What's up with the analog "echo" phenomenon?

I was recently listening to an Analogue Productions LP record: "The Intimate Art Pepper". I couldn't help but notice--particularly on the solo horn cuts--an anoying echo-type of phenomenon. Actually, it is more like a reverse echo with the quieter "decay" sound proceding the loud actual intended sound by about a half second. I hope this makes some sense. Actually, I've noticed this for years on other analog recordings. It is very noticable during very quiet passages just before a loud passage. I didn't expect to hear this on a supposed "audiophile" pressing.

Is this phenomenon a necessary evil in the analogue tape medium?