In the example on the left of the illustration there are ten different typefaces competing for attention. Can you see the two script fonts, three decorative fonts, two sans serifs, and three serifs? Mixing two script typefaces is rarely a good idea.
Varying the size of just two or three typefaces would have been more effective with, perhaps, a single script typeface for the pull-quote and initial cap.
The eight typefaces in the second example break two different rules — too many fonts and using fonts that are too similar in style. There are four different sans serif typefaces and four serif.
The lack of contrast, along with lack of consistency, throws off the design (more obvious when seen full size). There's little reason to use multiple typefaces if they are so similiar in style. It would probably be better to use that 'tried-and-true formula' of pairing one serif and one sans serif.
No hard and fast rule says you can't use five, six, or even twenty different fonts in one document. However, consistency and readability are important to good design and too many font changes can distract and confuse the reader. Make your font choices carefully and consider how many typefaces will be seen together — longer, multi-page publications, such as magazines, can often tolerate a greater variety of typefaces. For brochures, ads, and other short documents, limit typefaces to one, two, or three.