"I think there are reasons to be more and more bullish about Twitter, but we've got to see some of that proof," said trader Pete Najarian.
The most recent round of merger speculation—this time that Facebook could buy its smaller rival—sent Twitter shares nearly 6 percent higher to close above $36. Twitter co-founder Jack Dorsey will take the helm as interim CEO on Wednesday as current CEO Dick Costolo steps down.
Trader Brian Kelly said he would own Twitter with $33 to $34 per share as as stop. He would sell it once a new CEO is announced, as that decreases the likelihood of a takeover and limits upside, he contended.
Trader Guy Adami agreed, noting that Twitter rose from a similar trading range in December.
Apple launched its streaming service Apple Music on Tuesday, entering an already crowded space. Its shares rose 0.7 percent on the day, closing above $125.
Adami said investors should hold the stock with $122 per share as a stop. Kelly instead looked to streaming competitor Pandora, which has shed nearly 50 percent in the last year to settle around $15 per share.
"I'd much rather try to buy that at $15 than I would Apple," he said.
Karen is long BABA, BAC, C, FINL, FL, GOOG, GOOGL, JPM, KORS, M, TACO and URI. She is short SPY. Her firm is long ANTM, AAPL, BAC, C, DIS, DRI, FBT, FINL, FL, GOOG, GOOGL, GPS, IBB, JPM, KORS, M, SUNE, URI, XBI, KORS call spreads, URI calls and SPY puts. Her firm is short IWM, SPY and MDY. Karen Finerman is on the board of GrafTech International.
Long AMAT, AAPL, BABA, BAC, BMY, BP, CSX, DISCA, DKS, FOXA, GE, KKR, KO, LLY, MRK, PEP and PFE. He is long calls AAPL, ABX, BAC, BBY, C, DAL, ETFC, FCAU, GS, HYS, INVN, JPM, LULU, NUAN, OC, PNR, S, SPY, SXC, SYY, UAL, UBS, USB, VOYA, VZ, WYNN, XLF and ZIOP. Today, he bought SPY calls and WYNN calls. Today, he sold DE calls.
Brian Kelly is long BBRY, BTC=; TAN and TSL. He is short euro, yuan and yen.
Guy Adami is long CELG, EXAS and INTC. Guy Adami's wife, Linda Snow, works at Merck.