I can tell you from personal experience that excluding evidence at a trial because the chain of custody is weak is rare, certainly the exception rather than the rule.
The typical situation where the chain is not particularly strong is for the trial judge to nevertheless admit the evidence, ruling that the weakness of the chain goes only to "the weight of the evidence [i.e., how much weight or credence the jury will give it], not its admissibility".
" -- Vincent Bugliosi; Page 442 of "Reclaiming History" (Endnotes)
JohnM