Probably the problem is not on the ATX 12v connector, there are security fixtures on the motherboard that stop the computer to work if there is something wrong, usually a short circuit or memory not correctly placed.
Best way to find where the problem is is disconnecting everything but the cpu, power supply (connect atx 12v) and pcspeaker, and try to connect stuff one by one:
1 - nothing
pcspeaker should beep (you can read beep codes in your motherboard's manual) NOMEMORY ERROR
If no beeps then you have a short-circuit/power supply problem -> try with another power supply
- memory
If beep code is the same as previously then memory is not installed properly, actually this is a common mistake, you should place it in a way that both place holders "click", if you don't hear the "click" do it again
- graphics card
If your motherboard has on-board graphics you dont need to place graphics card to have a BIOS screen, if it doesn't place your graphics card, now you should have a BIOS screen when turning on the computer.
If all successful until now, connect the Hard Drive, you should be able to boot
If all successful until now connect everything else.
When one of the steps fail, just repeat the install/connection, it is easy to place memory or graphics card wrongly.
Also, I have seen some computers not turning on when inside the case, something was touching the back of the motherboard causing a short-circuit, that might also be the case, in last case, build the computer outside the case, you can turn it on by short circuiting the pins where the PWR connection is on your motherboard.
Hope this will help.