Minimum no. of grouping variables in mixed models I have data where I collected y and predictors x1, x2, x3, x4 over multiple years at county-level. county is nested within 'State' and 'State' is nested within 'Province'
My model is 
lmer(y ~ x1 + x2 + x3 + (year|Province/State))

Is there a rule of thumb that says how many State should be present under Province for using lmer? For e.g. no. of State within a Province looks like this:
provinceID      stateCount
1                  1
2                  1  
3                  2
4                  3   
5                  5     
6                  6
7                  8
8                  9     
9                  11
10                 16  
11                 17
12                 20  
13                 30
14                 37

 A: 
My model is
lmer(y ~ x1 + x2 + x3 + (year|State/Province))


If State is nested within Province then you have specified the nesting the wrong way around. The correct formulation for the random effects is:
(year|Province/State) 

which expands to
(year|Province) + (year|Province:State)

As for the number of states per province, I think you should be fine, however, in the case of province ID's 1 and 2, you might consider combining them into a single province.
Doug Bates, the original primary author of lme4 fits nested models with 2 lower units per upper unit, in the Pastes dataset here (subsection 2.1.3.1).
To directly answer the question in the OP: I don't believe there is a rule of thumb, but if there is, then following blindly would be very unwise. This issue isn't really a problem because when fitting nested random effects, the software will estimate a variance for the interaction between the lower and upper level factors (as you can see from the expansion of State/Province) so provided that the total number of these is sufficient, which in your case it clearly is, there should not be a problem.
