The blockchain is already 15 GiB. Bitcoin can't replace credit cards if we don't solve this problem.
Read Satoshi's original paper, chapter 7:
https://bitcoin.org/bitcoin.pdfExplains how to prune the blockchain, to about 5MB/year. (removing transaction hashes, only keeping the merkle root hash in the end)
Now we just have to wait for devs to implement it!
Bandwidth is another thing.. Come-from-beyond has a good point, but if internet speeds keep rising, I personally think by the time BTC gets to 10000transactions/s this will be high enough.
A lot of home connections are already able to easily keep up with that speed.