The term "Cronbach's alpha" itself is the wrong name giving false information about the formula's original author. I will discuss history later and concentrate on mathematical properties here.
Alpha is commonly known to have values between zero and one. Some textbooks explain that the value of alpha ranges between 0 and 1, and some do not explain the range of alpha at all. Literature that explains that alpha can be negative is rare.
The notion that 'alpha has a value between zero and one' does not mean 'it is rare that alpha has a value less than zero', but 'it is impossible for alpha to have a value less than zero'.
I challenge the impossible. Let's do a simple analysis in SPSS for proof. The first variable is filled with random numbers. The second variable copies the first variable. The third variable multiplies the first variable by -1. The following is an example.
Below is the SPSS analysis process and results. Unlike what is commonly known, we got a value of -3.
Although the above example is artificial, the actual data analysis sometimes reports that alpha is negative. For example, a negative alpha value may be calculated if the score of the item measured by the inverse item is not reversed (e.g., 7-> 1, 6-> 2, ..., 1-> 7).
If you misunderstood that alpha always has a value greater than zero, you might have thought of the following:
(a) Reliability has a value between 0 and 1.
(b) Alpha equals reliability.
(c) Therefore, alpha has a value between 0 and 1.
The wrong premise here is 'alpha equals reliability'. Alpha is an estimator, or estimate, of reliability. If alpha's prerequisites are violated, alpha is not equal to reliability. Alpha may have a negative value if the prerequisites violate severely. The prerequisite for alpha, that is being "tau-equivalent", will be described later.
Not all reliability coefficients have this problem. Structural equation model-based reliability coefficients (e.g., composite reliability) always have a value between 0 and 1.
For additional reference, please see:
Cho, E., & Kim, S. (2015). Cronbach's coefficient alpha: Well known but poorly understood. Organizational Research Methods, 18, 207-230.