There is no agreed threshold. If you are doing a regression-type model, and the missing value is a RHS variable, then one common work-around is to add 'missing' as a separate variable.
More specifically, if your missing variable is categorical, you will be adding in dummy variables for each category. You can then just have another category 'missing'.
If the missing varialbe is continuous, then you can code missing values to some arbitrary value (eg zero) and also include an additional dummy variable equal to one if the variable is missing.