There were great Vaiyaakaranaas (not just grammarians but Linguists) before and after Sage Panini. Sage Panini himself refers about 16 Vaiyaakaranaas (linguists) in his book Ashtadyayi (some are also referred by Sage Yaska the etymologist who lived before Sage Panini). Sage Panini borrowed some of their rules to build Ashtadyayi - The greatest linguistic canon in existence. There were sure other Vaiyaakaranaas whose works are lost and also Sage Panini hadn't referred /used in Ashtadyayi. Names of these linguists who were referred by Sage Panini (partial) are:
Apishaali, Audumbaraayana, Chakravarma, Gaargya, Galava, Kaasakritsna, Kasyapa, Paushkarasaadi, Shaakalya, Shaakataayana, Shaunaka, Sphotaayana, Vaarshayani, Vaarthaaksha, Vaajapyaayana, Vyaadi, and the Etymologist Yaska
Can we say that all those 16 Vaiyaakaranaas (linguists) who Sage Panini referred were neighbors of Sage Panini and were living in the same time? - It would be silly to say like that - but some western scholars and so called "Indian rationalists" say that or mean that in an indirect way.
First, western Indologists have fixed the time of Sage Panini to 2500 years back or around 500 BC. (The rationale behind fixing this timeframe is not properly established). This date fixing was done during 19th Century during British rule with very limited data and very little understanding of Sanskrit. Because Buddha conveyed his message in Paali the colloquial dialect of Eastern India spoken that time – Paali was chosen, so that the message not only reaches the educated elite (Sanskrit scholars) but also the uneducated masses - thus it is very evident that the widespread scholarly language used at that time was Sanskrit. If so, then it must be much older than Buddha and a scholarly language must have a tight grammar – thus the Grammar of Sanskrit must be much older. In my view Sage Patanjali and his linguistic cannon Mahabhashyam must have existed before Buddha’s /Mahavira’s time – this is evident from the fact that Jaina texts of Mahavira and Parswanatha discussions didn’t have any non-Paniniya usage (apaniniya prayoga) where as the Ramayana and Mahabharata and many puranas have many non-Paniniya usage.
Secondly some Indologists keep writing that Sage Panini invented Sanskrit language, etc. without any basis or research. Ashtadyayi, the linguistic canon written by Sage Panini was descriptive and not Prescriptive in those days. - only after the days of Buddha when scholars embraced Buddhism and started writing in Paali it has become Prescriptive - so it is unwise to say Sage Panini Structured the language etc. - the structure (grammar) was existing before - Sage Panini structured the Grammar Rules in an easy-to-read manner in a small book having 4000 formulas (3959 to be precise). In those days Ashtadyayi was much easier in comparison with other grammar texts or Pratishakyam (vedic grammar) texts.
Thirdly some argue that Sanskrit wasn’t a spoken language Sage Patanjali’s Mahabhashyam explains how the usage of Sanskrit was in various regions. He highlights the differences of same verb /noun usage with different meanings in different parts of Ancient India.
Those 16 Vaiyaakaranaas (linguists) who Sage Panini referred must have lived at least 100s of years before Sage Panini if not more. Because since we are reading the texts of Sage Panini now - after 2500 years (this timeframe is again as per western Indologists). So it could be possible that Sage Panini was reading the texts of earlier Vaiyaakaranaas (linguists) who lived 1000 years before Sage Panini. More over the works of earlier linguists were spread in many volumes and also they were having regional grammatical flavors and possibly some outdated usages of Sanskrit. Finally to provide an easy way of understanding the structure of the language, and instead of having to refer many works, Sage Panini wrote a treatise in which all the rules of the language were codified in a simple manner - thus born Ashtadyayi.
Most importantly those 16 Vaiyaakaranaas (linguists) and their schools referred by Sage Panini were different from the "Nava-Vyakarana" (9 grammatical traditions) - referred in Valmiki Ramayana (Sri. Hanumaan is a Navavyakaranavettaa - a scholar of all the nine grammar schools). (The 9 grammar schools are Aindra, Kaumaara, Shaakta, Saaraswata, Chandra, Soorya, Braahma, etc.). Some of the Indian scholars themselves confuse between the 9 Vyakarana schools (which are Devataa or God’s schools) and the pre-Paninian 16 Vyakarana schools, which are the grammar traditions of various regions /various times of Ancient Bharata (India) and not that of Devataa – both these 2 groups are different.
After Sage Panini, Sage Katyayana in 300 BC (this timeframe is again as per western Indological theories) added 23,000 new words - in linguistics parlance these many words take over 100s of years to get added to the language - provided the language has in-built word generation capabilities - Morphological capabilities. Sage Katyayana also added few missing rules to Ashtadyayi as the language and its usage has transformed from the time of Sage Panini - this itself proves that there is a long gap between these 2 linguists.
Later in 200 BC (this timeframe is again as per western Indologists) Sage Patanjali in his explanation treatise of Ashtadyayi called Mahabhashyam added another 28,000 new words due to the usage patterns and transformation of the language - this proves that a]. Sanskrit was widely used, b]. there existed a long gap between the times of Sage Panini and Sage Patanjali. These facts are known to Sanskrit scholars of Vyakarana - it is a pity that still many choose to tread the lines of western indological theories either because of no point in fighting with people who do surface level research and fix timeframe for Sanskrit or out of indifference. Which ever way this is injustice to the language and to our forefathers. I'm not writing this so that we all can feel proud that the language is much older, than what it was thought of, but to do justice to this great language. No point in simply talking about Sanskrit without putting it to use. We have responsibility to learn Sanskrit deeply and unlock the secrets hidden in millions of Sanskrit scientific treatises - still many of then are in Palm-leaf /wooden Manuscript forms.
Great Vaiyaakaranaas (linguists) like Bartrhari, Battoji Dikshita, Narayana Battathiri, Kaunta Bhatta, Nagesa Bhatta are Post Panini/Katyayana /Patanjali – just to quote few names. Each one of these and many other great Linguists have contributed many things to the Sanskrit linguistic science. Eg:. Semantics, Psycholinguistics, Neuro-Linguistics, etc. were dealt in detail in 5th Century AD itself by Sage Bartrhari in his work Vakyapadiyam.
Since Vyakarana (grammar) is a Vedanga (part of Veda) like Veda and the Sanskrit language, Vyakarana is also Anaadi (time immemorial). So when we talk /quote about Sanskrit Language we need to keep all these in Mind. Some myopic views do exist that Sanskrit was born in 1500 BC and not before, etc. We as learned should know how to brush aside the untruth.
Thus with all these we can assume that Sage Panini could have lived before 4000 years back, not later – After the period of rebuilding of the Vedic civilizations during the start of Kali yuga and after the deluge due to which the Dwaraka City /state submerged in the ocean – 5114 years back. These dates are debated in Indian Science Congress and some are proven (accepted by majority of scientists) based on planetary positions and astronomical calendar systems. – some info http://en.wikipedia.org/wiki/Kurukshetra_War and http://articles.timesofindia.indiatimes.com/2007-03-10/special-report/27883505_1_mahabharata-ramayana-epics ; on Dwaraka http://www.youtube.com/watch?v=zeDMSXOhDbY