Can I use Eevee's "Call for Family" to search through my deck if I already have four Eevee in play or in the discard pile?
You can use the attack, but you cannot search your deck since it is public knowledge that you won't find an Eevee because all four copies of the card are already in known areas.
Call for Family
Source: PUI Rules Team (2008-07-17); TPCi Rules Team (2023-03-23)Mar 23, 2023Compendium → Am I allowed to use Gossifleur's "Call for Family" attack if my Bench is already full?
You can use the attack, but since it's public knowledge that you can't put any Pokemon onto your Bench the entire effect of the attack fails; you can't even search your deck.
Call for Family
Source: Supreme Victors FAQ (2009-08-27), PUI Rules Team (2009-08-27), PUI Rules Team (2020-05-07)May 7, 2020Compendium → Emolga with a Grass energy attached is poisoned and uses its "Call For Family" attack to get Virizion EX with "Verdant Wind" ability. Does it take poison damage or not?
No, because Virizion EX is in play before the end of the turn and Poison damage is taken between turns.
Call for FamilyVerdant Wind
Source: TPCi Rules Team (2014-01-02), TPCi Rules Team (2014-05-08)May 8, 2014Compendium → If I use Fantina and then my Flying Pikachu V with Big Parasol attached uses its "Fly" attack and successfully lands heads, would my opponent's Giratina V's "Shred" attack do any damage to Flying Pikachu V or not?
Shred isn't affected by "any effects on the Defending Pokemon"; so the effects from Fantina and Big Parasol and Fly would all be ignored, and Shred would do damage.
FlyShredBig ParasolFantina
Source: TPCi Rules Team (2022-09-29)Sep 29, 2022Compendium →