Can you put two commanders into the command zone if they have the Partner ability?

Partner is a keyword ability introduced specifically for the Commander format. Under the rules for Commander (and codified in CR 702.124), if a legendary creature has Partner, it can be paired with another legendary creature that also has Partner. Both creatures then serve as your commanders simultaneously.

Both commanders begin the game in the command zone together (CR 903.3a). Your deck is built around both of them: your deck's color identity is the combined color identity of both commanders, and you may cast either one from the command zone at any time you could normally cast a creature spell, paying the commander tax separately for each.

The commander tax (an additional {2} for each time that commander has been cast from the command zone, per CR 903.8) is tracked independently for each partner commander. Casting one does not increase the tax for the other.

If either commander would be put into a graveyard, exile, hand, or library from anywhere, its owner may instead return it to the command zone (CR 903.9). This applies to each partner independently.

Example: You choose Tana, the Bloodsower and Tymna the Weaver as your two commanders. Both have Partner. They both start in the command zone. Your deck's color identity is Red, Green, White, and Black. You can cast Tana for {3}{R}{G} on turn one and later cast Tymna for {1}{W}{B} — each tracks its own commander tax separately.