By bitcoin network becoming too big, I honestly don't understand what you mean because if we are talking about bitcoin's network, it is already very massive like someone else have mentioned here, you can know this if or when you want to run your own node and need to download the full bitcoin blockchain from the very beginning to the very present, you will have to have a computer with no less than 2 terabytes of data space so that after you are done with the download, you will still have some space left so that the machine can run optimally.

But if what you mean is if the bitcoin network become too congested, like many millions of people are transaction on the bitcoin blockchain at the very same time, well, what this will do is that it will definitely lead to the network becoming congested like we have seen on some occasions in the past, this congestion will lead to hiked transaction fees and hiked time one will have to wait to get his or her transaction confirmed.