Yes, it is absolutely possible to overtrain. I was just reading about a study in Cycling magazine where participants biked hard for three days straight, and their performance (I think measured in terms of amount of watts/power output) decreased by A LOT on the third day (maybe 30%??? I can't remember the exact numbers but it was significant). The good news is that they were back to max output after 48 hours of rest.
I would recommend taking a day or two off. Gym is hard on the joints so little aches and pains don't go away as easily if you train everyday. Also, when you condition the muscles get mini-tears in them which take 48-72 hours to heal. It's the healing of the muscle that makes it stronger so if you're conditioning the same muscle every day and not allowing the time to heal, you won't be getting stronger. This is why it's recommended that weight lifting be done every three days.
I've also found that when the body is physically tired, sleep requirements increase, but I'm guessing none of you are actually sleeping more so you'll feel sleepy-tired, too. This is going to make you not as mentally alert.
So all this physical and mental tiredness will make you more prone to injuries, more prone to silly mistakes because you're not 100% able to focus which can lead to fluke accidents, and you won't be making the most out of your workouts.
Taking a day or two off will help recharge you mentally and physically. It will also not hurt your performance. Anyone training for a major event (5k run, marathon, triathlon, etc) knows to start tapering on training in the last week. Your physically fitness is there - you won't gain much in the last week - but to perform on race day your body has to be rested. Take two days, and I bet you'll come back stronger than ever, more mentally focused, and re-energized and motivated to work hard.
