Amici lecythum et vas unguentarium tenent ne cadant neve frangantur.

They are the direct objects of tenent (they hold).

• lecythum = a lekythos / oil flask in the accusative singular
• vas unguentarium = a perfume jar / ointment vessel in the accusative singular

Since the friends are holding these things, Latin puts them in the accusative.

Because amici is the subject of tenent.

• amici = friends
• nominative plural = the friends are the ones doing the action

The verb tenent is also plural, which matches amici.

Tenent is the 3rd person plural present active indicative of tenere.

So it breaks down like this:

• ten- = the verb stem
• -ent = they in the present tense for this verb pattern

So amici ... tenent = the friends hold ...

Because ne here introduces a negative purpose clause.

So the sense is:

• tenent = they hold
• ne cadant neve frangantur = so that they do not fall and are not broken

In Latin, purpose clauses normally use the subjunctive.
That is why you see:

• cadant instead of an indicative form
• frangantur instead of an indicative form

Not quite. Here ne does not simply mean ordinary not. It means lest or so that ... not.

So:

• ne cadant = lest they fall / so that they do not fall

This is different from non, which is the usual word for straightforward negation.

Neve means and not, nor, or and that not.

So the pattern is:

• ne cadant = lest they fall
• neve frangantur = nor be broken

It links a second negative idea to the first one.
You can think of neve as ne + and.

A very literal sense would be:

• They hold them so that they may neither fall nor be broken.

Because the two singular objects are joined by et and treated together as a plural idea.

• lecythum et vas unguentarium = the flask and the perfume jar
• together, they are two things
• so Latin refers back to them with plural verbs: cadant and frangantur

In English we do the same thing:

• The cup and the bowl are on the table, not is on the table.

The objects.

Even though Latin does not repeat the nouns, the meaning is understood from context:

• The friends hold the flask and the perfume jar
• so that those objects do not fall or get broken

So cadant and frangantur refer back to lecythum et vas unguentarium, not to amici.

Because the ideas are slightly different:

• cadant = they may fall
  Falling is expressed actively: the objects themselves fall.

• frangantur = they may be broken
  Breaking is expressed passively: the objects receive the action of being broken.

This is very natural in both Latin and English:

• so that they do not fall and are not broken

Latin could have used a different phrasing, but this combination is perfectly normal.

Frangantur is 3rd person plural present passive subjunctive of frangere (to break).

You can identify it like this:

• frang- = verb stem
• -antur = 3rd person plural passive ending in the present subjunctive

So it means:

• they may be broken
• or in smoother English here, be broken

Cadant is the 3rd person plural present active subjunctive of cadere (to fall).

So:

• cad- = verb stem
• -ant = 3rd person plural subjunctive active for this conjugation pattern

In this sentence it means:

• that they may fall
• or more naturally, that they fall
• with ne: that they not fall

Because that would just state a fact:

• non cadunt = they are not falling
• non franguntur = they are not being broken

But this sentence is not mainly stating a fact. It is expressing the purpose of holding them:

• They hold them so that they will not fall or be broken.

For that reason Latin uses ne + subjunctive, not non + indicative.

Because Latin adjective placement is flexible.

• vas unguentarium = an ointment vessel / a perfume jar

Placing the adjective after the noun is very common in Latin, especially when the adjective is simply describing the noun in a straightforward way.

So this word order is normal and does not need a special explanation.

Literally it means something like an ointment vessel or a perfume container.

• vas = vessel, container
• unguentarium = for ointment / perfumed oil

So the phrase describes a container used for perfume or scented oil.

It is a loanword from Greek.

A lekythos was a kind of small flask, often for oil or perfume. Latin borrowed the word, and in this sentence lecythum is the accusative singular form.

So a learner may notice that it looks a bit unusual because it is not a native Latin word of the most common type.

Because Latin word order is much freer than English word order.

The sentence is arranged like this:

• Amici = subject
• lecythum et vas unguentarium = objects
• tenent = main verb
• ne cadant neve frangantur = purpose clause

This is a very natural Latin structure.
English usually depends heavily on word order, but Latin relies more on word endings, so the writer has more freedom in arranging the sentence.

Not quite. That is too literal in a way that sounds un-English.

A better literal translation would be:

• The friends hold the flask and the perfume jar lest they fall and lest they be broken
• or lest they fall and not be broken is wrong, because neve means nor / and not

The best natural English is something like:

• The friends hold the flask and the perfume jar so that they do not fall or get broken.
• The friends are holding the flask and the perfume jar so they won’t fall or be broken.

So it is helpful to understand the Latin structure, but not to copy it too mechanically into English.