If you have more predictors in your regression than you have values to predict you will get a perfect prediction model.
In other words: If you have 10 measurements, and 10 variables that could be related to these measurements, you can fit a model that perfectly explains the variation in the measured variables. This is called overfitting. In practice, only 1 or 2 variables really explain what is going on. The rest is just 'filling the gaps'
I would recommend looking at stepwise regression if you do not know which of the 1000 predictors in X are the ones you need. The check the p-value for the predictor to see how likely it is that predictor is just a random variable.