I applied for the Premier Rewards Gold card with 10K bonus on 4/29. After I got the card, I sent secure messages four times, call them three times asking for the 75K bonus (Bonus ID: 5894), and all got shot down. I also asked for the 50K bonus targeting to business school student and got a firmly no. So I gave up. I thought they probably knew I am a not a big spender. My only other Amex card is the Costco Trueearnings card with only $2000 limit.
In late June after I hit $1000 spend on the gold card, I gave another try and sent a secure message but asking for only the 25K offer this time. Surprisingly, they approved and added 25K points to my account.
Yesterday, I got greedy and sent a message asking again for the 75K bonus with another Bonus ID: 5985. Today I got the reply saying another 50K have been added to my account, and they appreciated my membership since 2005. I am happy now.
So I'd say, don't give up if you got shot down initially, and keep trying after you spend $500 or $1000 on the card.