Are 4 cores enough

Solution
You do not say what resolution your screen is. A 1920x1080 system will need less (graphics) horsepower than a 4K system. Running games mostly depends on the graphics card, no so much the CPU. However, if the CPU is old or slow, then that can impact gaming, too. Your system has no such problem. If you decide to upgrade, remember to buy a 6th or 7th generation core-i7. However, you will only get hyperthreading extra, probably not worth it. If you really want to upgrade you will have to buy an 8th Gen (6 cores), or 9th Gen (8 cores) CPU. I would say it is not worth doing that yet (you would also need a new motherboard for 8th or 9th gen). Wait until you experience problems in a particular game. Even then, upgrading the graphics card would...
It depends on what the CPU's architecture and platform is, if you're referring to Core 2 Quad's, then most probably no...besides games are not entirely dependent on the CPU, you'll need a platform to back that CPU in order to play the game. Mind sharing your specs like so:
CPU:
Motherboard:
Ram:
SSD/HDD:
GPU:
PSU:
Chassis:
OS:
 
You do not say what resolution your screen is. A 1920x1080 system will need less (graphics) horsepower than a 4K system. Running games mostly depends on the graphics card, no so much the CPU. However, if the CPU is old or slow, then that can impact gaming, too. Your system has no such problem. If you decide to upgrade, remember to buy a 6th or 7th generation core-i7. However, you will only get hyperthreading extra, probably not worth it. If you really want to upgrade you will have to buy an 8th Gen (6 cores), or 9th Gen (8 cores) CPU. I would say it is not worth doing that yet (you would also need a new motherboard for 8th or 9th gen). Wait until you experience problems in a particular game. Even then, upgrading the graphics card would probably work better than upgrading the CPU.
 
Solution
When it comes to cores: right now you are fine if you have 4 cores. But games developers are going to start more and more to scale for more cores. If you buying/building a complete new system that is intended to serve you for many years to come I would look into 6 or 8 core (coffee lake or ryzen). But you will not need more than 8 cores for gaming. Battlefield1 and 5 for example definitely appreciate more than 4 cores (but will still run on 4 cores).
 
BF5 is phenomenally CPU intensive at least in the beta. I'm at 1440p with a 1070 and i7 7700k, I am getting 70% useage or more on all threads constantly, if I had a 1080 ti I would be tapping out a 7700k and at 1440p where more load should be on the CPU. I'm very curious to see how it will handle a quad core non hyperthreaded i5 at 1080p with something like an RX 580.
 
Well, it will be definitely possible to play with 4 cores. The minimum specs for Battlefield are:

Battlefield V open beta PC minimum system requirements
OS: 64-bit Windows 7, Windows 8.1, and Windows 10
Processor: AMD FX-6350 or Intel Core i5 6600K
Memory: 8GB RAM
Graphics card: AMD Radeon HD 7850 2GB or Nvidia GeForce GTX 660 2GB
DirectX: 11.0 Compatible video card or equivalent
Online connection requirements: 512kbps or faster Internet connection
Battlefield V open beta PC recommended systems requirements
OS: 64-bit Windows 10 or later
Processor: AMD FX 8350 Wraith or Intel Core i7 4790
Memory: 16GB RAM
Graphics card: AMD Radeon RX 480 4GB or Nvidia GeForce GTX 1060 3GB
DirectX: 11.1 Compatible video card or equivalent
Online connection requirements: 512 KBPS or faster Internet connection