I guess the most important thing is that the samples in your data are well spread, because no matter how much data you have, more data would always be better.
After all, if you try to learn to distinguish between cat and dog pictures, you can't expect your model to perform well if you only feed it cat images.
As suggested in the answer by Kevin L, it makes sense to consider the difference between training error and testing error. If your testing data is independent from your training data, this gives an indication as to how well your model generalises to the data that is unavailable.
Something I would like to add to that is the fact that a large difference between training and testing error only tells you that your model does not generalise well, i.e. you are overfitting on the training data. More data will probably help, because now the network also needs to model the extra data points, hence cannot overfit that much anymore. However, it might be more worthwhile to change your model such that it generalises better. This chapter from an excellent book explains what types of regularisation exist and how they can be applied in networks to get better generalisation.
If you were looking for a more quantitive measure, I recently found this question on quora. It is about an auto-encoder, but I guess it should also be applicable to your example. I have no idea whether this is correct (please let me know), but I would reason that for instance for MNIST, one could argue that you try to reduce images with a maximum of 28 * 28 * 8 * 10 000 = 62 720 000 bits entropy to ten classes in one-hot encoding with 10 * 10 * 10 000 = 1 000 000 bits of entropy. Because we are only interested in the 1 000 000 bits of entropy at the output, we can say that with 1 000 000 parameters, each parameter represents a single bit, which is 1e-4 bit per sample. This means you would need more data. Or you have too much parameters, because e.g. with 100 parameters, you have 10 000 bits per parameter and therefore 1 bit per sample. However, I would like to emphasise once again that it is the first time I have seen something like this and if anyone could confirm this, I would be grateful.