What do the characters of Cordelia and the Fool have in common?

In Shakespeare's King Lear, Lear's youngest daughter, Cordelia, and Lear's Fool are the resident truth tellers. Both of them speak truth to power (that is, to Lear), and they also speak truth to just about everybody else in the play.
Many of the characters in King Lear are devious and duplicitous, either by their own nature or by circumstance (e.g., Goneril, Regan, Cornwall, Edmond, Oswald), but Cordelia and the Fool aren't afraid to tell the truth—they both seem incapable of not telling the truth—and they're both willing to face the consequences for doing so.
Cordelia speaks the truth in the first scene of the play, and the consequence for her failing to be hypocritical and flatter and fawn over her father (as her sisters, Goneril and Regan, do), is banishment without an inheritance, land, or a dowry. Telling the truth ultimately leads to her death.
Likewise, Lear threatens the Fool with whipping almost daily. The Fool speaks the truth even in response to being threatened with whipping.

KING LEAR: An you lie, sirrah, we'll have you whipped.
FOOL: I marvel what kin thou and thy daughters are: they'll have me whipped for speaking true, thou'lt have me whipped for lying; and sometimes I am whipped for holding my peace. I had rather be any kind o' thing than a Fool: and yet I would not be thee, nuncle . . . (1.4.172–176)

Another thing that Cordelia and the Fool have in common: they don't appear in the play at the same time. They're never on stage together. This leads some Shakespeare scholars to speculate that the roles of Cordelia and the Fool were played by the same actor—keeping in mind that in Shakespeare's time female characters were played by male actors, either men or boys.
It wouldn't be unusual for the same actor to play both characters. Actors often doubled roles (played more than one part) in Shakespeare's plays for the simple reason that there were often more characters in a play (up to thirty or more) than the number of actors in Shakespeare's acting company (usually about sixteen actors). Just about every character, except the lead characters, was doubled.
Another theory is that Cordelia actually acts the part of the Fool in the play in order to stay close to Lear and care for him even after she's been banished.
It's generally accepted, however, that Robert Armin (c. 1563–1615), a well-known comic actor and a member of Shakespeare's acting company, played the Fool in King Lear. In fact, some scholars believe that Shakespeare wrote the character of the Fool with Armin in mind.
If Armin played the Fool, it's doubtful that he also doubled the role of Cordelia, but it's not out of the question that another actor might play both parts.