This is actually my favorite theory, and there's an overwhelming amount of evidence that lends itself towards this being true. In order for it to make sense, however, I would go even deeper and include that Caroline and GLaDOS are basically the exact same person, and by extension, GLaDOS is Chell's mother.

Firstly, I don't believe that GLaDOS actually deleted Caroline at the end of Portal 2. She's a liar, for one thing, and what we see of her relationship with Caroline seems to make deleting her impossible. Caroline is very heavily implied to be part of the core programming of GLaDOS, as shown when GLaDOS is put into the potato battery and her more Caroline-like sides begin to appear for the first time, indicating that the most basic parts of her personality like with Caroline. She empathizes with Cave Johnson in a way only Caroline seemingly could have, such as agreeing vehemently with him in the lemon speech and how, the first time we hear Caroline's recorded voice, GLaDOS actually absentmindedly synchronizes her own voice to the recording. And besides, surely Caroline MUST be part of the foundation of the GLaDOS system, because GLaDOS actually stands for Genetic Lifeform and Disc Operating System, implying that some biological being is the source of her intelligence, presumably Caroline. Without Caroline, GLaDOS is nothing but a computer system with no human emotions, and she's clearly still just as emotional and cold at the end of the game after deleting Caroline.

Additionally, it's hinted throughout the game that Caroline and Cave had more than just a professional relationship. When we hear Caroline's voice, she sounds very enthusiastic, which isn't that realistic for a secretary working in a failing research company. Also, GLaDOS is very excited by the lemon speech that Johnson makes, which probably comes from her emotional "Caroline" side. And, as an easter egg, the player can find a portrait of Johnson and Caroline together, and GLaDOS even comments that she recognizes it, though she's clearly having trouble summoning the memory. All these things point to Johnson and Caroline being in a possible romantic relationship, and it's not too much of a far cry for them to have gone on to have a child, especially because, as GLaDOS's memories slowly come back, she seems to become warmer and more receptive to Chell, who she's theoretically remembering is her daughter. There's even a cut line in which she seems to remember Chell's name and compliment her: "You really are doing great... Chell.".

You may be wondering why, then, GLaDOS acts so icily towards Chell and tries to kill her so often. Simple: Caroline was made into GLaDOS unwillingly (given by one of Johnson's last words ordering the scientists to put Caroline "in his computer" if they have to after his death, whether she wants go or not), and forgot her past as part of the process. She killed everyone in the facility in spite and started burning through test subjects left in stasis, and Chell happened to be on the list. Having forgotten her daughter, she treats Chell like any other subject and gets mad when Chell outsmarts her. Evidence for her transition to GLaDOS being unwilling exists in a series of cut lines like this one, where she's telling Johnson not she "does not want" something, though it's unclear in the line what. Presumably, Johnson, in his dying, broken mind, forced Caroline into the computer. (A darker interpretation here is that he raped her, impregnating her with Chell. While the cut lines are rather rapey, and that rapiness is in fact why Johnson's voice actor refused to record his side of the conversation, (this is apparently not true, my mistake) I personally don't believe that. It seems to me that, as GLaDOS began to remember more about herself, she would feel disgust for him rather than excited appreciation.)

The single most damning piece of evidence, however, is the turret opera at the end of the game. GLaDOS claims to have deleted Caroline, but still chooses not to kill Chell, because killing her "is hard." She then sends Chell away through the elevator, where she finds herself cornered by turrets who could presumably be ordered by GLaDOS (who claims to hate Chell at this point) to shoot and kill her. Instead of doing so, however, they begin to sing an opera song in Italian. The turret opera is a beautiful ending to Portal 2, but it gets especially interesting when you translate the lyrics. This page includes the full translated lyrics of the song, which is performed by GLaDOS's voice actress through beings presumably controlled by GLaDOS, by the way. The turrets (and by extension, GLaDOS) actually call Chell "mia bambina," or "my child," and warn her "stay away, yes, far away from science" (most likely citing Aperture Science itself). GLaDOS doesn't want Chell locked up in such a dangerous facility where she's faced near death several times for the rest of her life. GLaDOS wants Chell to be free, and so she frees her, while also protecting her from the knowledge that a psychopathic murderous computer was her mother all along. GLaDOS, a mostly completely selfish character by nature, ends the game in a selfless act to help her daughter. And how does she finally send Chell off for the last time? By returning to her the one thing she felt she could call a companion.

Man, I love this game.

EDIT: To answer a few questions: The biggest reason I've seen here that this theory isn't true is that, in Chell's Easter egg science fair project she refers to "Dad" and "Mr. Johnson" as two separate people. To this, I respond that she was not wanted by Caroline and Johnson, and was thus put up for adoption and adopted by one of the other scientists in the facility. There's evidence for this in the fact that the portrait includes a little girl, presumably Chell, hiding around the edge of the frame, implying she wasn't considered part of the family. (Thanks to /u/risto1116 for pointing this out!) I personally believe she was adopted by Doug Rattman, as it would explain why, in the comic "Lab Rat" he knows her well enough to think she should be at the top of the testing list. Presumably, he knew she could beat GLaDOS.

Also, some people are pointing out that, in fact, Johnson's voice actor did not refuse to record the lines, and that's not why they were cut. I'm sorry about that, I was going purely off of memory, and I either remembered wrong or remembered an inaccurate source.