What is the downside of using four 16gb chips [in 4 slots]?
This is a good question, as there is no technical downside to having 4 x 4GB of RAM in a system. This comes down to the chipset, where if you were using Intel's latest X99 chipset, you'd have quad-channel RAM. So, using four sticks of RAM is what you want. But, if you had a dual-channel chipset, then using four sticks is also fine.
If your motherboard has four DIMMs max, which most do, and you're using a dual-channel capable chipset, then four sticks is fine. The only 'downside' to that would be your upgrade path, which would be maxed out at 16GB. You could use 2 x 8GB sticks, leaving two of them open to another 16GB, which would have your system enjoying 32GB of RAM.
So, to answer your question: no, there are no downsides - you're just limiting your system to 16GB of RAM versus 32GB. And unless you're doing heavy video editing or similar, 16GB of RAM is more than enough.