IMEI numbers being used to bust criminals?

Say a criminal had two SIM cards. One was used for legit work and to talk to family, but the other was used for nasty illegal activity, however both were used on the same single phone with a unique IMEI number. Could the police etc find out that calls made on both SIM cards were both made on the same phone which has the same IMEI number and then bust him? This is assuming the criminal couldn't afford a second phone or was too incompetent/lazy. Any thoughts?