I have heard a number of complaints recently that the chemicals (specificaly chloramine - a mixture of chlorine and ammonia) used in some water treatment facilities are causing the material the molded plastic refrigerator water reservoirs are made from to break down resulting in white or whitish particles appearing as flakes, flecks or specs in the dispensed chilled water. In that case the water filter is not usually the cause, instead replacing of the chilled water reservoir itself may be necessary.

Most manufacturers originally used a molded plastic water reservoir which is what seems to be effected and breaking down. Many fridge manufacturers are now selling a coil of plastic tubing (like that used in the rest of the refrigerator's water system) in place of a molded plastic reservoir. You can see some of the replacement kits at the links below.

Those makes/models that still only have the original molded plastic reservoirs available as a replacement may also be able to be replaced by one of the other manufacturer's/model's 'tubing kits' using the attachment fittings supplied with them.

  Reservoir Tubing Kits
Whirlpool fridge water reservoir replacement kit
Frigidaire water reservoir replacement kit
Maytag water reservoir replacement kit
Amana water reservoir replacement kit

  Tubing Connectors*
      - Quick connect -
Push Connector 1/4" OD to 1/4" OD
Push Connector 1/4" OD to 1/4" OD Elbow
Push Connector 1/4" OD to 5/16" OD
Push Connector 5/16" OD to 5/16" OD
      - Compression fitting -
Tubing Connector 1/4" OD to 1/4" OD
Tubing Connector 1/4" OD to 5/16" OD
Tubing Connector 5/16" OD to 5/16" OD
1/4" Tubing plus connector

* refrigeration tubing is measured in OD (outside diameter) rather than ID (inside diameter) as plumbing tubing is.

  Original Molded Plastic Water Reservoirs
Whirlpool molded plastic chilled water reservoir
GE molded plastic chilled water reservoir
GE molded plastic chilled water reservoir
GE molded plastic chilled water reservoir
GE molded plastic chilled water reservoir
Amana molded plastic chilled water reservoir
Maytag molded plastic chilled water reservoir

Frigidaire brands include Frigidaire, White Westinghouse, Tappan, Gibson, Gallery, Kelvinator and some Kenmore models.

Whirlpool brands include Whirlpool, Roper, KitchenAid, Estate, Whirlpool and many Kenmore models.

Maytag brands include Maytag, Magic Chef, Jenn Air, Hardwick and Admiral.

GE brands include RCA, Profile, Moffat, McClary, Hotpoint, General Electric and Beau*Mark.