Probably not, unless the high tier servant had a shit master.
I'm just spitballing here, but I think each Servant has a specific level of power they are meant to be at. If the master isn't up to the task, they are going to be weaker. An extremely powerful master might confer a bonus here and there (Illya granting Heracles life regens), but you probably can't take a servant too far above its baseline. The example I would use here is Saber Alter, who despite having Sakura's supply, isn't stronger then Rin!Saber beyond being able to use Excalibur Morgan more, or EMIYA, who has crap stats Rin or Hakuno.
Of course, between fights being more about compatibility then straight up powerlevels, powercreep saturating the upper tiers while leaving lower ones fairly barren (seriously, separating the servants requires something like B, B+, B++, A, A+, A++, EX at this point), and other hjinks expected of a grail war, the whole high tier v low tier is in itself a bad way to look at it except in cases where you have a clear gap, like Karna vs Astolfo or something.