Educated Guesswork

Umm... I know this is niggling and not really your point, but a tablespoon is 15 ml.

I didn't see the show in question to know the context, but I imagine that it depends on how fast you lose the blood. If it's fast enough, I would imagine a sudden 10% drop in blood volume could lead to funky reactions in your blood pressure, which could indeed cause you to pass out. Could also be that your body might do something funky to prevent additional blood loss by constricting vessels in the area you're losing blood from/diverting blood flow away from there, so if the blood loss were from the head or neck or something, I could picture physiological reactions which might make you pass out.

Some while ago, NPR had an item on "All Things Considered" about CSI, and how accurate it is or isn't (here it is: from 24 May 2004, Dose of Reality: 'CSI' vs. True Las Vegas Crime Lab). In it, they talked with a woman from the Las Vegas crime lab, who had been a consultant for the TV show. I haven't listened to it since, but, as I remember, they asked her how much of the things they do on the show can really be done, and she said that most of it can be done -- just not at the same place, at the same time, by the same lab, quickly. They'd have to be sending things out all over the country and waiting weeks to get the results back.

Since I expected that, what interested me more was her comment when asked whether it annoys them to see the liberties that are taken on the show when showing how the forensics people work and what they can actually do. She said, as I recall, that they weren't really bothered by it, per se, because they know that a drama show has to alter things for dramatic purposes (yeah, like having the labbies out there interviewing witnesses and confronting suspects). But what does worry them is when they see how juries often won't convict someone because evidence isn't there that CSI taught them to expect.