Same sort of thing happened to me roughly 3 weeks after I started back up after a couple years of relative inactivity.
I also consider myself one of those people who never seems to get sick and suddenly I had really bad sinus issues, and a possible fever. Even came down with impetigo (facial staph infection, wtf?) right after the congestion issues tapered off.
I simply decided to work through it all. There were a couple times during that period when I came home from work, ate and then crashed until I woke up just in time for work again the next morning, but more often than not I felt great in spite of my ailments, especially during and directly after a session in the gym.
I can also attribute much of that to my nutritional intake as well–another aspect that I had suddenly began paying attention to after several years of relative dietary apathy. I was definitely eating more and making better quality choices than before.
In the end, I don’t think my health issues were prolonged much longer (or at all) than if I had simply taken the time off.
I can definitely say that from a psychological standpoint I came out feeling like I was much further ahead, even if it probably didn’t matter that much from a physical strength/training progress standpoint.
I knew I was sick, but I just really didn’t feel like I was a particularly ill person–I couldn’t convince myself there was a real reason to stop except for a few symptoms that I didn’t feel the need to take any medication for anyways.
Maybe I got lucky… but I seriously doubt it. I’m pretty sure the vast majority of male human bodies in their 20’s and 30’s can handle mild forms of sickness and a beginner’s weightlifting regime at the same time and still progress, albeit in a less-than optimal fashion.
A critical failure at this point just doesn’t seem possible unless there are other factors at work, in which case, you probably already know why you shouldn’t be exerting yourself without some form of medical supervision.
So, as long as you don’t kid yourself into becoming a martyr, I’d pay attention to what your body is really telling you and weigh ALL of your symptoms carefully–like your energy level, disposition, etc.–and not just list the negative ones that hypochondriacs like to keep handy.
One month of lifting weights (I don’t know exactly what you were doing prior to or even during this time) seems, by definition, contraindicative of “overtraining” to me, but you’ll want to pay especially close attention to what your body’s really saying right now and play it day-by-day.
Besides, are you going to let a cold going to stop you in a couple of months from now if you catch one? Assuming you’ve also made appropriate adjustments to your nutrition and recovery, this setback is probably only due of the sudden influx of added stress you are now putting your body through. It will adapt accordingly.
Weightlifting isn’t AIDS.