Note that this is not currently the case in bitcoin, transaction fees have gone up with the current scarcity due to the arbitrary block limit, but the fees are still a pittance compared with block reward - which is the real incentive for mining (but will not always be the case as block reward reduces in the future).
This is somewhat of a sore point for the bitcoin community as a large (probably not majority, but large) portion of the user base / miners / nodes does not think scarcity of space is a good idea at current levels of transactions.
Also Ethereum (which at the moment uses a blockchain and mining mechanism very similar to bitcoins) does not impose a block size limit, rather leaves it up to the miners to decide on the "gas" limit (they have a computational limit rather than a block size, but it can be viewed as a parallel).