Volaran is correct: the specific overrides the general.
The Familiar section specifically states “For the purpose of effects related to number of Hit Dice, use the master’s character level or the familiar’s normal HD total, whichever is higher.”
All of the core sample familiars begin with less than a full hit die (unlike their Master who has a full 1d4) clearly indicating they CAN gain additional hit dice and since the Familiar section does not specify otherwise, they follow the general rules for Improving Monsters which states each hit die gained increases a magical beast’s Fort and Reflex saving throws by one, provides an additional (2+ Int mod) skill points, and may increase its base size (with associated changes).
Thus unless I missed an errata or Pathfinder seriously changed it, -ALL- Familiars have:
* a type of Magical Beast
* 10-sided Hit Dice
* a base attack bonus equal to their Master's BAB. (instead of their total hit dice)
* have half their Master's total hit points regardless of their hit dice.
* use the better of their own or Master's saving throws. As magical beasts, they have Good Fortitude and Reflex saves.
* Skill points equal to (2 + Int modifier, minimum 1) per Hit Die, with quadruple skill points for the first Hit Die. However they use the better of their own or Master’s skill ranks; applying their own modifiers.
* provide their Master with a specific ability (such as a skill bonus) while within 1-mile of him.
* gain additional abilities detailed within the Familiar section based upon their Master’s class level
* Darkvision out to 60 feet and low-light vision.
* Proficient with its natural weapons only.
* Proficient with no armor.
* Magical beasts eat, sleep, and breathe.
Therefore the question is not whether Familiars can improve, but to what degree. RAW clearly states they can earn/gain additional racial hit dice and templates. It likewise states the Master must take the Improved Familiar feat if/when a template is applied.
I suggest this feat be expanded to allow familiars to become Classed like other sentient creatures of comparable intelligence (6-15 depending upon their Master’s level), imposing a Class level-cap equal to their Master’s level-2 per the cohort rules to prevent the possibility of the familiar overshadowing its Master, allowing the familiar to optionally retain excess experience points unassigned until their Master levels further or be applied to additional Hit dice with the corresponding benefits.
For example Apprentice Harold summons a hawk to be his familiar, gain a +3 on his own listen checks whenever its within 1-mile. In exchange, the hawk transforms into a magical beast, gaining the standard abilities. As such, it uses its newly discovered abilities and racial skills to help Harold navigate a number of low-level situations, and help him avoid a number of what would have been overwhelming odds. Such events eventually compound into enough XP for the hawk to gain another hit die, improving its Listening skill by 2pts.
Eventually Harold reaches level 3 and selects the Improved Familiar feat to imbue his hawk with the Celestial template, providing it with Acid Resistance 5, Cold 5, Electricity 5; SR 7; and allowing it to use Smite Good once per day.
By this time, the hawk has gained sufficient Xp to increase another hit die, but instead decides (having an Int:7) to become a Rogue, applying its initial 32 skill points learning to fly (move) silently, search, and improve its Spot/Listen skills. It selects Improve Natural Attack for its first level feat to increase its talon damage from 1d4-2 to 1d6-2.
Harold spends the next several months scribing scrolls between rather mundane adventures, during which the hawk makes some lucky sneak-attacks earning enough Xp to advance further. However Harold has not, so rather than increasing its racial hit dice, the hawk opts to leave the points unassigned until Harold gains a level so it further its own career, gaining the rogue's evasion class ability.