In terms of database design, class design, and many other programming tasks, it is often useful to restrict your concept of numbers to there being only three numbers

That is a good rule of thumb, but it is just a starting point. The real rule is, in my mind, whether a data rule requires it; and if it does, how many. If how many cannot be answered with a specific number, a new (child) table is used for them. Otherwise, same table.

For example, if storing email addresses and one alternate, it may make sense to store both email addresses in the same table. Another example (from my current project), if you have an entity created with up to 12 parts (each part being similar to a level), it is likely better to store the 12 parts in the same table.

Technicalities, perhaps. But, this is what i do for a living.

If you want to search for entities which contain a "Foo" part then you need to do:

SELECT entity_id
FROM entities
WHERE part1='Foo'
OR part2='Foo'
OR part3='Foo'
...
OR part12='Foo';
[download]

Or perhaps slightly more concise:

SELECT entity_id
FROM entities
WHERE 'Foo' IN (part1, part2, part3, ..., part12);
[download]

I'd much prefer something like:

SELECT e.entity_id
FROM entities e
INNER JOIN entity_parts ep ON e.entity_id=ep.entity_id
WHERE ep.part='Foo'
[download]

When you get to searching for different combinations of parts, the advantages of joining become even more apparent:

SELECT e.entity_id
FROM entities e
INNER JOIN entity_parts ep1 ON e.entity_id=ep1.entity_id
INNER JOIN entity_parts ep2 ON e.entity_id=ep2.entity_id
WHERE ep1.part='Foo'
AND ep2.part IN ('Bar', 'Baz')
[download]

And in terms of the e-mail example, it's much easier to, say, enforce a UNIQUE constraint over a single column.

use Moops; class Cow :rw { has name => (default => 'Ermintrude') }; say Cow->new->name

When you get to searching for different combinations of parts, the advantages of joining become even more apparent

Problem is, that is much less efficient query. EAVs are relatively terrible on performance, requiring index/table scans aplenty, unless the RDBMS supports partitioning, which effectively turns it into a bunch of small tables.

And in terms of the e-mail example, it's much easier to, say, enforce a UNIQUE constraint over a single column.

Yes, that is simpler. But, per the example, why enforce uniqueness on primary and secondary addresses? That would preclude the use of a friend or relative as a backup.

Because they are attributes of a particular entity, they should each have their own column. Making two things like that unique, is not an intrinsic data rule, it is some form of business rule. A business rule should not redesign the data model to make it easier to enforce, that will only lead to problems upon problems later on. In this particular case, the rule can be enforced via a MATERIALIZED or INDEXED VIEW with a UNIQUE INDEX. (A TRIGGER could be used, but that'd be hiding code and, likely, terribly inefficient.)