Yes, chromium could be one.
Right from the start, you can guess that the unknown element must be a d-block element, or a transition metal.
That must be the case because in order to be able to accomodate six unpaired electrons you need to have access to the five d-orbitals transition metals have access to.
As you know, each orbital can hold a maximum of two electrons of different spin.
When degenerate atomic orbitals are filled, they follow Hund's Rule, which states that one electron is added to each degenerate orbital in a subshell before two electrons can be added to the same orbital in a subshell.
This implies that you can have a maximum of five unpaired electrons, one in each of the five d-orbitals.
So where will the sixth unpaied electron be placed?
Take a look at the electron configuration of manganese,
#"Mn: " ["Ar"] 3d^5 4s^2#
Manganese has five unpaired electrons in its five 3d-orbitals and two paired electrons in its 4s-orbital.
This is in fact a clue - you can have six unpaired electrons if you have five unpaired electrons in the 3d-orbitals and one unpaired electron in the 4s-orbital.
The element that matches this electron configuration is chromium,
#"Cr: " ["Ar"] 3d^5 4s^1#
You can rad more about why chromium's electron configuration looks like that here: