The stress cards are an urban legend.
From snopes.com:
Claim: Recruits in basic training are issued "stress cards," which when waved at demanding drill sergeants immediately entitle recruits to gentler treatment.
Status: False.
Examples:
[Collected on the Internet, 2002]
I have heard that during Clinton years, the Army issued new basic training recruits "Stress Cards" at select training bases as part of a test program. If the Drill Sergeant yelled too loudly or instructed the recruit to do anything that might cause them undue stress, the Private could simply hand the card to the Sergeant and they were to cease the offensive behavior.
[Collected on the Internet, 2000]
I read about them but they weren't using them when I was in. If I remember correctly, it was a little yellow card that they gave you. Apparently if things were getting tough for you in basic, you could flash the card and the DI would back off and give you a "break" so you could compose yourself. The standing joke was that the color of the card spoke for itself... The idea, if I remember right, was heavily criticized (and rightfully so, what are you going to do in real life when the bullets start to fly, pull out the stress card and hope the bad guys stop shooting at you?) and the idea was eventually canned.
[Collected on the Internet, 1999]
[The stress card is] a card these kids get when they go through basic training. when they are feeling overwhelmed or stressed out, they are to give it to their DI's (or whatever the call em now days) for a 'time out'.
my friend john, who was a 10 year enlisted veteran in the combat engineers who had done some 'hard time' in somalia (you remember the infamous fire fight over there involving the rangers) told me a story about a brand spanking new PFC who, during a field exercise, came up to john with this 'card' and said, "sgt, i need some time out. my stress card says im entitled to some time out." john, bless him, grabbed the stress card, tore it to bits and informed the kid, "stress card? you're in the REAL army now, kid. this is what i think of your &*^% stress card." Rip! Rip! Rip!
Origins: This is one of those tales that has the smallest kernel of truth to it, but that truth is almost unrecognizable in the form the scuttlebutt has since taken.
For a few years during the 1990s, the US Navy did issue "stress cards" to new recruits, but they weren't the "Get out of jail free" coupons military lore has since turned them into. Rather, these cards listed names and phone numbers of resources the newcomers could contact "if things pile[d] up." The cards were strictly for informational purposes: they informed recruits of available support services.
Navy trainers began reporting that some of the recruits had taken to raising their cards while being disciplined, as a way of signalling for time out. It's unclear whether any of those enduring basic training really thought that was the purpose of the cards or whether this was just standard armed forces jackassing, but the Navy took no chances and got rid of the cards.
This short-lived experiment with providing recruits with clear information about whom to contact when things went bump in the night has morphed into an unflattering and unsettling illustration of today's soldier as a creampuff. Notice how the story has mutated into one where the drill instructors are portrayed as honor bound to obey the cards when they are displayed to them, an aspect that wasn't part of things during the real cards' short life. The story has also widened its net; what was a Navy hand-out has, in the world of rumor, become a card issued to Army and Marine recruits, making this an Armed Forces-wide phenomenon.