Jewelry does not. Also, the starter island quests drop gear in other traits.

In this thread in post #2 of @code65536 , in the spoiler is an awesome list up of all quest items that are also an armor set piece incl traits.
https://forums.elderscrollsonline.com/en/discussion/296319/named-item-drops#latest

For Trainee the items are:
["Armor of the Trainee"] =
{
[1] = "{ 1115 }, -- Smuggler's Vest of the Trainee (Intricate)",
[2] = "{ 1119 }, -- Gallant Chain of the Trainee (Reinforced)",
[3] = "{ 7358 }, -- Jakarn's Machete of the Trainee (Sharpened)",
[4] = "{ 7368 }, -- Seyne's Bootknife of the Trainee (Sharpened)",
[5] = "{ 7369 }, -- Bonesplinter of the Trainee (Decisive)",
[6] = "{ 7392 }, -- Darj's Shield of the Trainee (Impenetrable)",
[7] = "{ 15567 }, -- Neramo's Staff of the Trainee (Powered)",
[8] = "{ 36573 }, -- Bloodthorn Dagger of the Trainee (Precise)",
[9] = "{ 36574 }, -- Kingsguard Hammer of the Trainee (Decisive)",
[10] = "{ 36575 }, -- Targoth's Shield of the Trainee (Reinforced)",
[11] = "{ 36581 }, -- Tazgol's Ancestral Axe of the Trainee (Charged)",
[12] = "{ 43574 }, -- Storm-Slave's Razor of the Trainee (Charged)",
[13] = "{ 43576 }, -- Hazak's Hauberk of the Trainee (Well-fitted)",
[14] = "{ 43577 }, -- Ratsplatter of the Trainee (Precise)",
[15] = "{ 43578 }, -- Sugar Jacket of the Trainee (Intricate)",
[16] = "{ 43579 }, -- Rootsplitter of the Trainee (Decisive)",
[17] = "{ 43580 }, -- Queen's Own Cuirass of the Trainee (Sturdy)",
[18] = "{ 43581 }, -- Broadhead Shield of the Trainee (Reinforced)",
[19] = "{ 43582 }, -- Mourning Bow of the Trainee (Ornate)",
[20] = "{ 44458 }, -- Earth-Turner Axe of the Trainee (Precise)",
[21] = "{ 44459 }, -- General Emax's Staff of the Trainee (Charged)",
[22] = "{ 44460 }, -- General Vette's Greatsword of the Trainee (Defending)"

Nic727 wrote: »

Does the ratsplatter of the trainee can drop outside training zone/quest rewards?
Same for other trainee set?

No the sets are specific to the zones

Nic727 wrote: »

Ok thx... But ratsplatter was a quest reward I think, but now its in a set... So if I deconstruct this item. I will have the possibility to get it again in the zone?

The named items that are quest rewards don't drop - they're just quest rewards.

Browiseth wrote: »

This set is like the bad joke answer to "we need a safe set for players to use earlygame"
>drops in tutorial zones
>simple bonuses
>all training

IDK if this set is a bad joke

It is never a set for a really pure build stacking everything all the way in DPS/HPS.
But that fits to the intended profile.

It is I think a very versatile set:
It comes in all weights and (therefore) the Jewelry comes in Arcane, Robust and Healthy.
You can use it very well as a solid platform to combine with another set in several directions of build set-up and experiment what direction fits you best.

I see still a use for it at CP 160 as easy going base set.

The training trait on two pieces of Armor does not hurt that much as long as you're not at CP cap.
And you can always chooce to skip the quests in the starter area's until you are at CP 160, to get better traits.

Its awesome, but hard drop , somebody has purple jewellry (arcane) ?