Within a game there's what is supposed to be a draw from a shuffled hand of five cards. In order to get a specific card with a certainty of 95%, I should draw log(1/20)/log(4/5) =~ 14 times. Let's say I've drawn 18 times, and still haven't seen the target card. At this point I begin to suspect that the presentation of randomly shuffled cards is either broken or disengenious.

How does one measure and test this system in order to come to the conclusion that it is no longer reasonable to claim it's working as advertised?

You would use Hypothesis Testing.

Here's a 38 page introduction to the subject... probably better than the summary on Wikipedia or on Mathworld...

http://www.sagepub.com/upm-data/40007_Chapter8.pdf

The probability of having 18 failures in a row with p(fail) = 0.8 is 0.8¹⁸ which is about 0.018 so you could reject the hypothesis that it is a fair draw at the 2% level.

EDIT: This page may be useful too

http://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair

In this specific instance if you are allowed to track each card as you select it, create a simple frequency distribution. Most of the time cards should track within a certain amount of variance.

Run your scenario 100 times. Each scenario is to select 14 cards. 4.3% of the time your event should happen.

You can automate to run a much larger amount of times. Following the law of large numbers, the value of your event happening should approach 4.3% of all trials.