Trumpet shank specifications
The 'shank' of a trumpet mouthpiece is critical to good mechanical connection between the mouthpiece and the instrument, affecting how the instrument feels and plays.
Two significant factors affect how a trumpet mouthpiece fits and plays with a particular instrument: Taper Angle and End Diameter
Taper Angle
The first critical aspect is the taper of the mouthpiece shank. The trumpet shank starts at a particular diameter and then slowly gets larger at a constant rate as you go up the mouthpiece.
In the case of most brass instrument mouthpieces, this taper is a "Morse #1 Machine Taper". This is a taper rate of change of 1:20.047 or an angle of 1° 25' 43" (or an "included angle" of 2.855°). This means for every inch you travel along the taper, the diameter will grow by 0.0498".
End Diameter
The end diameter of the trumpet shank helps determine how far into the receiver a trumpet mouthpiece will go before it seats solidly in the receiver.
In the case of Pickett Brass trumpet mouthpieces, the end diameter is 0.385". Manufactures vary by this diameter.

What effect do these factors have on the mouthpiece?
Gap problems
If the end of the mouthpiece is too small, the mouthpiece will go further into the instrument and could interfere and hit the end of the leadpipe. The mouthpiece will be loose and won't 'lock in.'
If the end of the mouthpiece is too large, then the mouthpiece will lock in and the gap between the end of the mouthpiece and the leadpipe may be too large, affecting how the instrument slots and sounds. Often a very large gap can make an instrument feel fuzzy or uncertain, unable to slot. If a the gap is overly small or non-existent, the instrument will not slot and can feel slippery.
This effect of "the gap" is important and can affect how an instrument responds and plays. Adjusting this gap through the use of Reeves Sleeves can allow a player to adjust and refine the gap distance.
Wobble problems
If the taper of the mouthpiece shank is incorrect and doesn't match that of the receiver, the mouthpiece will wobble in the receiver. It may feel like it locks in, but in fact it's only locking in on a very small section of the mouthpiece.
Many antique instruments have varying tapers and it can be difficult to find the right mouthpiece taper that locks into the receiver.