The original Likert scale was a 5-point scale. But there is no reason to limit yourself to precisely this number of points. You'll find everything from about 4 to 10 points on such scales (occasionally even more, though it's unlikely we gain additional information by expanding beyond 7 or so).
The most important thing to bear in mind is that odd-numbered forms always have a "neutral" response option (the middle) while even-numbered forms force the respondent to make a choice one way or the other. Each approach has its advantages.
One thing to consider is the type of statistical analyses that you intend to carry out. For items with ordered categories response format, it is common to treat them as continuous interval level measures, so long at the number of categories is about 5 or more. So it is not unusual to use procedures such as regression analysis, ANOVA, or factor analysis with responses to 5 - point Likert type scales. But you are more likely to get criticism from reviewers, when using such methods with only 4 response options. Unless you are going to use more advanced procedures such as ordinal regression, my experience is that reviewers or journal editors are likely to ask that you treat responses as categorical, which would limit you options available, particularly if your sample is small, or if there are a small number of responses to some of the categories.
NONE AT ALL, except to remember -at the time of statistical treatment- that the scores of the responses to the items, despite the fact that some "believe or agree that they do", ARE NOT OR ARE AT AN INTERVAL LEVEL, YES NO ORDINAL
Likert scale is also subjected to a lot of debates and controversies in regards with the analysis and inclusion of points on the scale. Auseful article pl readArticle Likert Scale: Explored and Explained
I am obviously very late in responding to this question (my bad). You have some very good responses to the question you raised. When using a four point scale you will be loosing variance in your responses. A five point scale gives you more variance which can be important in terms of the statistical analyses done. John Crawford makes this point in his response above. Several people have also raised the issue of a neutral response choice within the scale. A likert scale is a forced choice assessment , you are forcing people to choose from the options that you give them. What if the respondent does not find any of the choices that represent their thinking? This may encourage some respondents to leave the item blank. In this situation you may provide a situation where the scale is no longer useful, or you have a scale that is missing input on one item. You lose some of the variance in your responses. Some scholars would suggest that you make those respondents think, do the cognitive work and make a choice. When you force respondents to make a choice from the list you have provided are you "creating" a response rather than measuring the construct. Forcing respondents to choose among the choices you have provided you are forcing the respondent to make a choice that is not reflective of the respondent's thinking. In doing this are you not "creating" rather than measuring the respondent's thinking? In providing this neutral response I think it important that you assign a score of 0 to the neutral response. If you give the neutral item a number rather than 0, your total scale score is higher than it should be. Hope this is helpful. Ed