You're talking about this?http://web.media.mit.edu/~leah/LilyPad/build/turn_signal_jacket.html
The LEDs are all in parallel.
The LEDs have a voltage drop, say 2V for Red since you using a 3V supply:https://www.sparkfun.com/datasheets/DevTools/LilyPad/Q150OVS4.pdf
The current per LED will then be a max of: (3V -2V)/100 ohm = 10mA
Plus there is the resistance of the thread, and the voltage out of the arduino will be less than 3V since the current is going thru a transistor.
The 28 ohm/LED junction will never be lower than 2V, so max current thru the 28 ohm resistor is (3V -2V)/28 = 35mA, and it gets shared among the LEDs, so 5mA/LED with 7 LEDs. Above 20mA, the Arduino output will droop, so the voltage at the 28 ohm/LED junction will likely be lower, See Section 29 of the datasheet:
VOH Output High Voltage( Note 3). Condition: IOH = -10mA, VCC = 3V
Minimum output at 85C: 2.3V
Sub in 2.3V for 3V above, and the current available drops even more.
If the coin cell was actually good for 250mAH, you would not have very long life from 1 battery.