If you're getting the same amount of protein every day, it doesn't matter whether you drink a shake before bed or at another time — you'll gain the same amount of muscle and strength.
Evidence from Studies
Supporting (5)
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Neither pre-sleep nor post-exercise protein consumption influences resistance exercise training adaptations in older adults
Daytime and nighttime casein supplements similarly increase muscle size and strength in response to resistance training earlier in the day: a preliminary investigation
Muscle-Related Effect of Whey Protein and Vitamin D3 Supplementation Provided before or after Bedtime in Males Undergoing Resistance Training
Protein Supplementation after Exercise and before Sleep Does Not Further Augment Muscle Mass and Strength Gains during Resistance Exercise Training in Active Older Men.
Casein Protein Supplementation in Trained Men and Women: Morning versus Evening
Contradicting (1)
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Effects of pre-sleep protein consumption on muscle-related outcomes - A systematic review.
Gold Standard Evidence Needed
According to GRADE and EBM methodology, here is what ideal scientific evidence would look like to definitively prove or disprove this specific claim, ordered from strongest to weakest evidence.