Is it possible to be genuinely mentally unstable and coldly calculating simultaneously?

I am looking for genuine insight here to help me better understand and deal with a situation at my office. I work with a person who claims to be mentally ill and suicidal. She appears to be unable to conceal her struggles and is often out of the office. When she does come in, she shuts herself in her office crying or walks around making a fairly open display of her distress with red eyes and sniffling. She is very theatrical about her situation and makes it hard to ignore. She has called out saying that she has been hospitalized several times, but has never presented any proof of this that I know of. She sometimes will abruptly leave work because she is “so upset” and depart in a hysterical, tearful state. At those times she gives the appearance of being out of control.

On the other hand there are some ways in which she seems extremely calculating and manipulative. For example, she is in charge of our payroll, which must be submitted to the company that actually issues the funds every other Wednesday. Frequently, Monday is the day she flees the office in hysterics, but before she runs out she will turn in the payroll, two days early, without marking herself absent for the rest of the day even though she knows she is preparing to leave. Then she will not show up for the rest of the week, but she’s already turned in payroll through to Friday, paying herself for 40 hours in a week where she might have worked 4. She also doesn’t deduct the time that she is out from her accrued leave or adjust her hours on the next payroll to correct the overpayment. If you are out though, she will deduct your leave time (or dock you if you don’t have any) faster as you can blink and always seems to know the status of everyone else’s leave time to the hour. She will take paid vacation days when she shouldn’t have any vacation time left. She comes in late, takes long lunches, sits in her office with the door shut making personal calls while others do her work, and takes a minimum of 3 half hour smoke breaks a day. This has been going on for the 2.5 years I’ve been working there, although it started gradually and has gotten increasingly worse to the point that it is now impossible to ignore. Our employers are truly convinced that she is suicidal and they are afraid to say or do anything for fear she will kill herself and it will be “their fault”. My boss has even openly said to me, “We’ve decided to just pay her for showing up whether or not she does anything.” I don’t know whether she is really having problems or the whole thing is a charade, but all of this raises my question, which is:

Is it really possible for someone who is genuinely mentally unstable and emotionally distraught to, at the same time, be so scheming? I’m having a hard time understanding how that can be and how a person could claim to be unable to function at work yet seemingly be so coolly calculating. The two things seem so diametrically opposed. If she’s really ill, I want to be able to feel empathy for her, but I’m having a hard time believing it. Sometimes I feel like she’s full of it, but sometimes I think I’m just being mean and maybe I don’t know. Can you help me understand? Can those two opposing aspects really co-exist?